Summary: The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions are obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The technique used is the well-known iteration process of S. Kaniel and M. Shinbrot [Commun. Math. Phys. 58, 65–84 (1978; Zbl 0371.76061)].