Various existence theorems are given for the generalized Enskog equation in R^3 and in a bounded spatial domain with periodic boundary conditions. A very general form of the geometric factor Y is allowed, including an explicit space, velocity, and time dependence. The method is based on the existence of a Liapunov functional, an analog of the H-function in the Boltzmann equation, and utilizes a weak compactness argument in L^1.