The classical theory of rigid bodies systems undergoing perfect bilateral constraints has received firm mathematical foundations for a long time. We extend it to the case where there are perfect unilateral constraints also. The formulation follows the line initiated by Schatzman (1978) and Moreau (1983). We give a parametrization-free formulation and try to identify the most general form of impact constitutive equation which is compatible with well-posedness. Then, well-posedness is proved under the assumption of analyticity of the data, since it is known, thanks to Bressan (1960) and Schatzman (1978), that uniqueness does not hold in the C∞ setting.