The Kac ring model is studied rigorously. Considered as an isolated system, its description is given in classical and quantum mechanics; the definition of the quantum states is made precise. The most interesting results are: (1) In the classical model, except for pathological cases, equilibrium is obtained independently of the initial state in the thermodynamic limit of a probabilistic theory; (2) as long as the number of sites and polarizations is finite, the master equation does not apply; (3) the explicit form of the non-Markovian terms in the evolution equation is obtained; (4) the memory of the initial state disappears from the evolution equation of the diagonal part of the density operator for a large class of nontrivial scattering operators. Finally, several unsolved problems are mentioned.