In 1867, just two years after laying the foundations of electromagnetism, J. Clerk Maxwell presented
a fundamental paper on kinetic gas theory, in which he described the evolution of the gas in terms
of certain ‘moments’ of its velocity distribution function. This inspired Ludwig Boltzmann to
formulate his famous kinetic equation, from which followed the H -theorem and the connection with
entropy. On the occasion of the 150th anniversary of publication of Maxwell’s paper, we review the
Maxwell–Boltzmann formalism and discuss how its generality and adaptability enable it to play a key
role in describing the behaviour of a variety of systems of current interest, in both gaseous and
condensed matter, and in modern-day physics and technologies which Maxwell and Boltzmann could not
possibly have foreseen. In particular, we illustrate the relevance and applicability of Maxwell’s
formalism to the dynamic field of plasma-wakefield acceleration.