The sweeping process, introduced some time ago by the author with motivation in plasticity theory, today remains an object of mathematical research. It is considered in this paper as the prototype of an evolution conditioned by inequality constraints. Since the governing differential requirements are only of order one with respect to time, this provides a simplified setting for analysing some numerical and theoretical features also present in unilateral dynamics. The latter is governed by differential inclusions of order two, for the numerical handling of which the existing literature proposes diverse strategies, briefly discussed. The paper is especially intended to offer an introduction to the numerical approach called 'contact dynamics'.