This lecture constitutes a short introduction to the subject of the stability (in the sense of Lyapounov) of evolution variational inequalities and non-smooth dynamical systems. After recalling some basic concepts and results in Convex and Set-Valued Analysis and in the field of Differential Inclusions (including sweeping processes), we present some basic existence results and an Invariance principle for a class of first order evolution variational inequalities. Using this approach, stability and asymptotic properties of important classes of second order dynamic systems with dry friction are studied. Using Brouwer's topological degree, necessary conditions for asymptotic stability of evolution variational inequalities are also studied.