Maxwell's fundamental studies in the period 1859--1879 on molecular physics and allied subjects are considered. His two very important papers of 1859 and 1866 on the dynamical theory of gases are analyzed, as are his two last extensive papers of 1879 on transport phenomena in rarefied gases and on statistical mechanics. Other papers, notes, lectures, and his book on the theory of heat are also considered. Particular attention is devoted to Maxwell's pioneering contribution to the development of statistical methods and his deep understanding of the statistical origin of the second law of thermodynamics. The connection between the work of Maxwell and that of Clausius and Boltzmann is discussed. It is emphasized that the development by Boltzmann of statistical methods was based to a large degree on the pioneering work of Maxwell. The influence of Maxwell's ideas on the development of the theory of inhomogeneous gases in the present century is also demonstrated.