The Anderson model is a paradigm for quantum disordered physical systems. Originally conceived in a condensed matter context, it is however difficult to study experimentally in solids. The quantum kicked rotor which has also a paradigmatic status for classical and quantum Hamiltonian chaotic systems, can be easily realized using laser-cooled atoms in a standing wave. Somewhat surprisingly, there is a mathematical mapping between these two systems, and the latter can be used as a "quantum simulator" for the former. In this talk, I will describe a series of experiments using the atomic kicked rotor allowing a rather complete characterization of the Anderson physics, and in particular a precise determination of the critical exponent of the Anderson metal-insulator transition.