In 1959 Mark Kac introduced a simple model for the evolution of a gas of hard spheres undergoing elastic collisions. The main simplification consisted in replacing deterministic collisions with random Poisson distributed collisions.

It is possible to obtain many interesting results for this simplified dynamics, like estimates on the rate of convergence to equilibrium and validity of the Boltzmann equation. The price paid is that this system has no space structure.

I will review some classical results on the Kac model and report on an attempt to reintroduce some form of space structure and non-equilibrium evolution in a way that preserve the mathematical tractability of the system.