This thesis is concerned with the consideration of the H-theorem in a statistical manner and the information that may be derived from it as to the variation with time of an isolated mechanical system, and especially the approach to equilibrium. A historical introduction is given in which it shown how the need for such a statistical approach arose, and hoy the question of the behaviour of the fluctuations about the values of H predicted by the unrestricted H-theorem became important. The type of behaviour suggested by the Ehrenfests is quoted and to verify this it is found to be necessary to consider in detail actual models. Two classical models, the urn model and the wind-wood model, are considered, and then the procedure is generalized so as to include the whole class of models of the type consisting of two groups of particles, the one group moving and interacting with the members of the second group which are fixed. The transition probabilities and the rate of change of H, and the mean time of recurrence of a fluctuation are found for these models by considering the influence of fluctuations upon the Stosszahlansatz values for the numbers of collisions. The results confirm the postulates of the Ehrenfests. In assumptions common to the statistical treatment of collision processes.