The paper is devoted to mechanical systems with a finite number of degrees of freedom. After showing how inequality requirements in evolution problems can be handled through differential inclusions, one introduces dynamics by an elementary example of unilateral mechanical constraint. Then a general setting is constructed for multibody multicontact systems. The description of unilateral interaction at each possible contact point is formalized, with account of possible friction. This generates the numerical time-stepping policy called Contact Dynamics. The treatment of collisions or other frictional catastrophes in this framework leads to measure-differential inclusions, an essential tool in nonsmooth dynamics. The energy balance of nonsmooth evolutions is discussed. Two illustrations of the proposed numerical methods are presented. The former concerns the mechanisms of collapse of a bridge arch under local forcing. In the latter, the construction of a conical pile of grains is simulated, in order to investigate stresses in the bulk and the distribution of pressure on ground.