The **Fields Medal** is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The Fields Medal is sometimes viewed as the highest honor a mathematician can receive. The Fields Medal and the Abel Prize have often been described as the mathematician's "Nobel Prize". The Fields Medal differs from the Abel in view of the age restriction mentioned above.

The prize comes with a monetary award, which since 2006 has been C$15,000 (in Canadian dollars). The colloquial name is in honour of Canadian mathematician John Charles Fields. Fields was instrumental in establishing the award, designing the medal itself, and funding the monetary component.

The medal was first awarded in 1936 to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions.

The Fields Medal is often described as the "Nobel Prize of Mathematics" and for a long time was regarded as the most prestigious award in the field of mathematics. However, in contrast to the Nobel Prize, the Fields Medal is awarded only every four years. The Fields Medal also has an age limit: a recipient must be under age 40 on 1 January of the year in which the medal is awarded. This is similar to restrictions applicable to the Clark Medal in economics. The under-40 rule is based on Fields' desire that "while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others."

The monetary award is much lower than the 8,000,000 Swedish kronor (roughly 1,400,000 Canadian dollars) given with each Nobel prize as of 2014. Other major awards in mathematics, such as the Abel Prize and the Chern Medal, have larger monetary prizes, comparable to the Nobel.

In 1954, Jean-Pierre Serre became the youngest winner of the Fields Medal, at 27. He still retains this distinction.

In 1966, Alexander Grothendieck boycotted the ICM, held in Moscow, to protest Soviet military actions taking place in Eastern Europe. Léon Motchane, founder and director of the Institut des Hautes Études Scientifiques attended and accepted Grothendieck's Fields Medal on his behalf.

In 1970, Sergei Novikov, because of restrictions placed on him by the Soviet government, was unable to travel to the congress in Nice to receive his medal.

In 1978, Grigory Margulis, because of restrictions placed on him by the Soviet government, was unable to travel to the congress in Helsinki to receive his medal. The award was accepted on his behalf by Jacques Tits, who said in his address: "I cannot but express my deep disappointment — no doubt shared by many people here — in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration."

In 1982, the congress was due to be held in Warsaw but had to be rescheduled to the next year, because of martial law introduced in Poland on 13 December 1981. The awards were announced at the ninth General Assembly of the IMU earlier in the year and awarded at the 1983 Warsaw congress.

In 1990, Edward Witten became the first physicist to win this award.

In 1998, at the ICM, Andrew Wiles was presented by the chair of the Fields Medal Committee, Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized Fields Medal". Accounts of this award frequently make reference that at the time of the award Wiles was over the age limit for the Fields medal. Although Wiles was slightly over the age limit in 1994, he was thought to be a favorite to win the medal; however, a gap (later resolved by Taylor and Wiles) in the proof was found in 1993.

In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal and did not attend the congress.

In 2014, Maryam Mirzakhani became the first woman as well as the first Iranian, Artur Avila the first South American and Manjul Bhargava the first person of Indian origins to win the Fields Medal.

This is a list of the universities that have graduated Fields medalists. It only includes those institutions that have graduated two or more medalists. See List of Fields Medal winners by university affiliation for complete affiliation.

*École Normale Supérieure is a combination of schools (in its anglo-saxon definition) representing the elite of the scientifique community in France.
**The unified University of Paris no longer exists since its dissolution in 1969. Its successor universities (numbered from 1 to 13) are here taken into consideration globally.
This is a list of the universities that Fields medalists have been affiliated to at the time the prize was awarded. It only includes those institutions that have had two or more medalist affiliates. See List of Fields Medal winners by university affiliation for complete affiliation.

*This institution is now known as the University of Lorraine.
The medal was designed by Canadian sculptor R. Tait McKenzie.

On the obverse is Archimedes and a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world). The date is written in Roman numerals and contains an error ("MCNXXXIII" rather than "MCMXXXIII").
On the reverse is the inscription (in Latin):
CONGREGATI
EX TOTO ORBE
MATHEMATICI
OB SCRIPTA INSIGNIA
TRIBUERE
Translation: "Mathematicians gathered from the entire world have awarded [understood 'this prize'] for outstanding writings."

In the background, there is the representation of Archimedes' tomb, with the carving illustrating his theorem On the Sphere and Cylinder, behind a branch. (This is the mathematical result of which Archimedes was reportedly most proud: Given a sphere and a circumscribed cylinder of the same height and diameter, the ratio between their volumes is equal to ⅔.)

The rim bears the name of the prizewinner.