We conduct a detailed study of the existence theory for nonlinear hemivariational inequalities of second order. The problems under consideration are strongly nonlinear and not necessarily of variational nature. So we employ a variety of tools in order to solve them. More precisely, we use the general theory of nonlinear operators of monotone type, the method of upper-lower solutions, the multivalued Leray-Schauder principle, nonsmooth critical point theory coupled with Landesman-Lazer conditions and linking techniques and also truncation and penalization techniques. The problems that we examine involve Dirichlet boundary conditions; in the last section we also examine a problem with a nonhomogeneous and nonlinear Neumann boundary condition.