The Lagrangian points (pronounced [ləˈgɹɒɲ.dʒi.ən] or [laˈgʀɑ̃.ʒjɑ̃]); (also Lagrange point, L-point, or libration point), are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects (such as a satellite with respect to the Earth and Moon). The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to rotate with them. They are analogous to geosynchronous orbits in that they allow an object to be in a "fixed" position in space rather than an orbit in which its relative position changes continuously.
A more precise but technical definition is that the Lagrangian points are the stationary solutions of the circular restricted three-body problem. For example, given two massive bodies in circular orbits around their common center of mass, there are five positions in space where a third body, of comparatively negligible mass, could be placed which would then maintain its position relative to the two massive bodies. As seen in a rotating reference frame with the same period as the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the centrifugal force are in balance at the Lagrangian points, allowing the third body to be stationary with respect to the first two bodies.

History and concepts
The five Lagrangian points are labeled and defined as follows:

The Lagrangian points
The L₁ point lies on the line defined by the two large masses M₁ and M₂, and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attractions of the two other objects effectively cancel each other out.
Example: An object which orbits the Sun more closely than the Earth would normally have a shorter orbital period than the Earth, but that ignores the effect of the Earth's own gravitational pull. If the object is directly between the Earth and the Sun, then the effect of the Earth's gravity is to weaken the force pulling the object towards the Sun, and therefore increase the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L₁ point, the orbital period of the object becomes exactly equal to the Earth's orbital period.
The Sun–Earth L₁ is ideal for making observations of the Sun. Objects here are never shadowed by the Earth or the Moon. The Solar and Heliospheric Observatory (SOHO) is stationed in a Halo orbit at L₁, and the Advanced Composition Explorer (ACE) is in a Lissajous orbit, also at the L₁ point. The Earth–Moon L₁ allows easy access to lunar and earth orbits with minimal change in velocity, and would be ideal for a half-way manned space station intended to help transport cargo and personnel to the Moon and back.

L1
The L₂ point lies on the line defined by the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal force on the smaller mass.
Example: On the side of the Earth away from the Sun, the orbital period of an object would normally be greater than that of the Earth. The extra pull of the Earth's gravity decreases the orbital period of the object, and at the L₂ point that orbital period becomes equal to the Earth's.
The Sun–Earth L₂ is a good spot for space-based observatories. Because an object around L₂ will maintain the same orientation with respect to the Sun and Earth, shielding and calibration are much simpler. The Wilkinson Microwave Anisotropy Probe is already in orbit around the Sun–Earth L₂. The future Herschel Space Observatory, Gaia probe, and James Webb Space Telescope will be placed at the Sun–Earth L₂. Earth–Moon L₂ would be a good location for a communications satellite covering the Moon's far side.
If the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁) then L₁ and L₂ are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:
where R is the distance between the two bodies.
This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M₂ in the absence of M₁, is that of M₂ around M₁, divided by .
Examples:

Sun and Earth: 1,500,000 km from the Earth
Earth and Moon: 61,500 km from the Moon L2
The L₃ point lies on the line defined by the two large masses, beyond the larger of the two.
Example: L₃ in the Sun–Earth system exists on the opposite side of the Sun, a little outside the Earth's orbit but slightly closer to the Sun than the Earth is. Here, the combined pull of the Earth and Sun again causes the object to orbit with the same period as the Earth. The Sun–Earth L₃ point was a popular place to put a "Counter-Earth" in pulp science fiction and comic books — though of course, once space based observation was possible via satellites and probes, it was shown to hold no such object. In actual fact, Sun–Earth L₃ is highly unstable, because the gravitational forces of the other planets outweigh that of the Earth (Venus, for example, comes within 0.3 AU of L₃ every 20 months).

L3
The L₄ and L₅ points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centres of the two masses, such that the point lies behind (L₅) or ahead of (L₄) the smaller mass with regard to its orbit around the larger mass.
The reason these points are in balance is that, at L₄ and L₅, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycentre of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycentre in the same ratio as for the two massive bodies. The barycentre being both the centre of mass and centre of rotation of the system, this resultant force is exactly that required to keep a body at the Lagrange point in orbital equilibrium with the rest of the system. (Indeed, the third body need not have negligible mass; the general triangular configuration was discovered by Lagrange in work on the 3-body problem.)
L₄ and L₅ are sometimes called triangular Lagrange points or Trojan points. The name Trojan points comes from the Trojan asteroids at the Sun–Jupiter L₄ and L₅ points, which themselves are named after characters from Homer's Iliad (the legendary siege of Troy). Asteroids at the L₄ point, which leads Jupiter, are referred to as the 'Greek camp', while at the L₅ point they are referred to as the 'Trojan camp'. These asteroids are (largely) named after characters from the respective sides of the war.
Examples:
The Sun–Earth L₄ and L₅ points lie 60° ahead of and 60° behind the Earth as it orbits the Sun. They contain interplanetary dust.
The Earth–Moon L₄ and L₅ points lie 60° ahead of and 60° behind the Moon as it orbits the Earth. They also contain interplanetary dust in what is called Kordylewski clouds.
The Sun–Jupiter L₄ and L₅ points are occupied by the Trojan asteroids.
Neptune has Trojan Kuiper Belt Objects at its L₄ and L₅ points.
Saturn's moon Tethys has two much smaller satellites at its L₄ and L₅ points named Telesto and Calypso, respectively.
Saturn's moon Dione has smaller moons Helene and Polydeuces at its L₄ and L₅ points, respectively.
The giant impact hypothesis suggests that an object named Theia formed at L₄ or L₅ and crashed into the Earth after its orbit destabilized, forming the moon. Stability
This section non-mathematically (intuitively) explains the five Lagrangian points using the Earth–Moon system.
Lagrangian points L₂ through L₅ only exist in rotating systems, as in the monthly orbiting of the Moon about the Earth. At these points, an outward (fictitious, as explained below) centrifugal force is balanced by the two radially attractive gravitational forces of the Moon and Earth.
Imagine using your hand to spin a weight at the end of a string. Your hand will feel an outward radial force in the direction of the weight. However, if you release the string, the weight does not travel radially outward in the direction of the sensed force. Instead, the weight travels tangentially to the circle, in the direction of spin at the time the string was released. This is why this force is called "fictitious". This same outward force is present in the Earth–Moon system, where the role of the string is played by the summed (or net) effect of the two attractive gravities, and the weight is an asteroid or science satellite. The Earth–Moon system rotates about a barycenter. Because the Earth is much heavier, this barycenter is located about a thousand miles below the Earth's surface, in the direction of the Moon (three thousand miles above the Earth's center, see Earth, Moon, and barycenter). Any object gravitationally held by the rotating Earth–Moon system will sense this (fictitious but calculably real) outward radial force away from the barycenter, in the same way your hand feels the outward pull of the string.
Unlike the other Lagrangian points, L₁ would exist even in a non-rotating (static or inertial) system. Rotation slightly pushes L₁ away from the (heavier) Earth towards the (lighter) Moon. L₁ is slightly unstable (see stability above) because drifting towards the Moon or Earth increases one gravitational attraction while decreasing the other, causing more drift.
At Lagrangian points L₂, L₃, L₄, and L₅, a satellite feels an outward centrifugal force, away from the barycenter, that exactly balances the attractive gravity of the Earth and Moon. L₂ and L₃ are slightly unstable because small changes in satellite position more strongly affect gravity than the balancing centrifugal force. Stability at L₄ and L₅ depends crucially on the satellite being pulled in three different directions, namely the outward centrifugal force away from the barycenter, balancing the inward gravitational forces towards the Moon and Earth.

Intuitive explanation
The Lagrangian point orbits have unique characteristics that have made them a good choice for performing some kinds of missions. NASA has operated a number of spacecraft in orbit around the Sun–Earth L₁ and L₂ points, including
ESA's Herschel Space Observatory (formerly called Far Infrared and Sub-millimetre Telescope or FIRST) in 2008 and the James Webb Space Telescope are also planned to be placed in orbit around L₂. ESA's Planck satellite planned for launch in 2008 will be placed in orbit around L₂.
The L5 Society was a precursor of the National Space Society, and promoted the possibility of establishing a colony and manufacturing facility in orbit around the L₄ or L₅ points in the Earth–Moon system (see Space colonization).
The Earth–Moon L₂ point has been proposed as a location for a communication satellite covering the far side of the Moon. NASA TN_D-4059
The Deep Space Climate Observatory was intended to be positioned at L₁, but has been put into indefinite storage after being built.

Lagrangian point missions
In the Sun–Jupiter system several thousand asteroids, collectively referred to as Trojan asteroids, are in orbits around the Sun–Jupiter L₄ and L₅ points. Recent observations suggest that the Sun–Neptune L₄ and L₅ points, known as the Neptune Trojans, may be very thickly populated, containing large bodies an order of magnitude more numerous than the Jupiter Trojans. Other bodies can be found in the Sun–Mars and Saturn–Saturnian satellite systems. There are no known large bodies in the Sun–Earth system's Trojan points, but clouds of dust surrounding the L₄ and L₅ points were discovered in the 1950s. Clouds of dust, called Kordylewski clouds, even fainter than the notoriously weak gegenschein, are also present in the L₄ and L₅ of the Earth–Moon system.
The Saturnian moon Tethys has two smaller moons in its L₄ and L₅ points, Telesto and Calypso. The Saturnian moon Dione also has two Lagrangian co-orbitals, Helene at its L₄ point and Polydeuces at L₅. The moons wander azimuthally about the Lagrangian points, with Polydeuces describing the largest deviations, moving up to 32 degrees away from the Saturn–Dione L₅ point. Tethys and Dione are hundreds of times more massive than their "escorts" (see the moons' articles for exact diameter figures; masses are not known in several cases), and Saturn is far more massive still, which makes the overall system stable.

Natural examples
The Earth's companion object 3753 Cruithne is in a relationship with the Earth which is somewhat Trojan-like, but different from a true Trojan. This asteroid occupies one of two regular solar orbits, one of them slightly smaller and faster than the Earth's orbit, and the other slightly larger and slower. The asteroid periodically alternates between these two orbits due to close encounters with Earth. When the asteroid is in the smaller, faster orbit and approaches the Earth, it loses orbital energy to the Earth and moves into the larger, slower orbit. It then falls farther and farther behind the Earth, and eventually Earth approaches it from the other direction. Then the asteroid gains orbital energy from the Earth, and moves back into the smaller orbit, thus beginning the cycle anew. The cycle has no noticeable impact on the length of the year, because Earth's mass is over 20 billion (2 × 10) times more than 3753 Cruithne.
Epimetheus and Janus, satellites of Saturn, have a similar relationship, though they are of similar masses and so actually exchange orbits with each other periodically. (Janus is roughly 4 times more massive, but still light enough for its orbit to be altered.) Another similar configuration is known as orbital resonance, in which orbiting bodies tend to have periods of a simple integer ratio, due to their interaction.

Other co-orbitals
The Lagrange points are mentioned in science fiction from time to time (most often hard science fiction), but, due to the general lack of public familiarity with them, they are rarely used as a plot device or reference.

In fiction

The L5 Lagrange point is mentioned in L5: First City in Space, an early IMAX 3D movie.
In William Gibson's novel Neuromancer, much of the action takes place in the L5 "archipelago", the location of many space stations.
The planet Troas in the stories "Sucker Bait" by Isaac Asimov and "Question and Answer" by Poul Anderson was located in the L₅ point of a fictional binary star system.
The space station Babylon 5 is described to be located "at the L-5 point in a binary star system between a moon and a barren, lifeless planet." [1]
In Hideo Kojima's video game Policenauts, the setting of the game, an O'Neill model space colony, is located at the L5 Lagrange point.
The genetically engineered humans or "Coordinators" of Mobile Suit Gundam Seed settle in L5 living in hour glass shaped colonies known as PLANTs L5

In Larry Niven's novels The Integral Trees and The Smoke Ring, the L4 and L5 points of the Smoke Ring (a ring of breathable air and plant life orbiting a neutron star) have become gathering places for masses of water, living things, and human settlers.
The eponymous interplanetary relay station in George O. Smith's "Venus Equilateral" stories was located in the L₄ point of the Sun–Venus system. L4

See also: Counter-Earth
In Peter F Hamilton's Night's Dawn Trilogy, a ZTT jump drive cannot be used in a strong gravitational field. In the first book of the trilogy, The Reality Dysfunction, the main characters cannot escape from a gas giant's gravity well before their pursuers catch up with them. Instead, they race to the Lagrange point between the gas giant and one of its moons in order to activate their drive. Successful execution of this untried and reckless maneuver gains captain Joshua Calvert the nickname "LaGrange" Calvert. In the second book The Neutronium Alchemist, a visit is paid to the supposed home planet of the Kiint, Jobis, which features three moons orbiting the Lagrange One point, rotating around a common centre.
In the third season of the TV series Lexx, the planets Fire and Water are found to reside in Earth's L3 point.
John Norman's Gor series takes place on a counter earth. L3

See also: Lilith (hypothetical moon)
In the TV series Quatermass II, the hostile aliens live on a small asteroid "no more than half a mile across" at a "theoretical point of equilibrium" on the dark side of the Earth, although neither L2 or Lagrange are mentioned by name (the term "Bieber Variation" is used instead).
In the Star Trek: The Next Generation episode, "The Survivors", the Enterprise is surprised by an enemy ship that had been hiding in a Lagrange point.
In the manga series Battle Angel Alita: Last Order, the ex-colony ship turned space station Leviathan 1 is at the L2 point in the Earth/Moon system.
In John Varleys book Wizard a religious group called the Coven set up a habitat at L2 to make themselves as remote as possible from the earth. L2 later slowly deteriorates into the space version of an unorganized shantytown as anyone with enough cash can set up home there. "L2 became known as Sargasso Point to the pilots who carefully avoided it; those who had to travel through it called it the Pinball Machine, and they didn't smile." L2

In Arthur Clarke and Stephen Baxter's novel Sunstorm, the L1 point plays a crucial role in the building of a shield that has the purpose of saving Earth from a storm of energy from the Sun.
In the Xbox video game Halo: Combat Evolved (2001) and sequel Halo 2 (2004), Halo Megastructures play key locations throughout the games. In Halo: CE and Halo 2, the Halo structures are in L1 Lagrange points between the Gas Giants (and a moon) Threshold and Substance, respectively.
In the Hugo Award-winning novel A Deepness in the Sky by Vernor Vinge, a temporary human habitat is built at the L1 point between the planet Arachna and its primary star, a highly variable dwarf called the On/Off Star. Unspecified Lagrange points