Maxwell was born in Edinburgh and studied at the university there (1847–50) and at Cambridge (1850–54), becoming a fellow in 1855. He was professor of natural philosophy at Marishal College, Aberdeen, from 1856 until 1860, when he became professor of natural philosophy and astronomy at King's College, London. He resigned in 1865 and worked on his estate in Scotland researching and writing. From 1871 he was professor of experimental physics at Cambridge.

Maxwell is regarded as one of the great physicists of the 19th century. At the age of 15 he produced a paper on methods of drawing oval curves. In 1857 he published a paper on the rings of Saturn, in which he analyzed the dynamics of the rings and proved that they could not be wholly solid or liquid. His own theory was that they were made up of many particles, and he showed that such a system would be stable.

Maxwell is regarded as one of the founders of the kinetic theory of gases – the calculation of the properties of a gas by assuming that it is composed of a large number of atoms (or molecules) in random motion. Maxwell, around 1860, put forward a statistical treatment of gases in Illustrations of Dynamical Theory of Gases. Maxwell and Ludwig Boltzmann obtained a formula for the way in which the speeds of molecules were distributed over the number of molecules – the Maxwell–Boltzmann distribution law. The kinetic theory of gases disposed of the idea of heat as a fluid (‘caloric’).

One interesting notion coming out of his work on the kinetic theory was the statistical interpretation of thermodynamics. A particular point was the idea of Maxwell's demon (1871) – a small hypothetical creature that could open or close a shutter between two compartments in a vessel, separating the fast molecules from the slow ones, and thus causing one part of the gas to become hotter and the other colder. The system would appear to violate the second law of thermodynamics. (In fact it does not; the gas decreases in entropy but there is an increase in entropy in the demon, using the idea that entropy is connected with ‘information’).

Maxwell's greatest work was his series of papers on the mathematical treatment of the lines of force introduced by Michael Faraday to visualize electromagnetic phenomena. He showed the connection between magnetism and electricity and demonstrated that oscillating electric charges would produce waves propagated through the electromagnetic field. He showed that the speed of such waves was similar to the experimentally determined speed of light, and concluded that light (and infrared and ultraviolet radiation) was in fact this electromagnetic wave. Maxwell went on to predict the existence of other forms of electromagnetic radiation with frequencies and wavelengths outside the infrared and ultraviolet regions. Heinrich Hertz first detected radio waves in 1888. Maxwell's theory was developed further by Hendrik Lorentz.

In Dynamical Theory of the Electric Field (1864) Maxwell put forward four famous differential equations (known simply as Maxwell's equations) describing the propagation of electromagnetic waves. The equations contain the speed of the waves c, a value that is independent of the velocity of the source. This was one of the facts that led Einstein to his special theory of relativity. Maxwell also wrote Treatise on Electricity and Magnetism (1873).