Inequality problems in mechanics can be divided into two main categories: that of variational inequalities which is concerned with nonsmooth and convex energy functionals (potentials), and that of hemivariational inequalities which is concerned with nonsmooth and nonconvex energy functional (superpotentials). While variational inequalities have been studied extensively, the study of hemivariational inequalities is more recent. Through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. In this talk, sample hemivariational inequalities in mechanics will be introduced and studied. Recent and new results will be reported on the numerical solution of hemivariational inequalities. Computer simulation results will be provided to show the performance of the numerical methods and to illustrate the numerical evidence of the theoretically predicted optimal convergence orders.