In old, dense stellar systems collisions of white dwarfs are a rather frequent phenomenon. We present the results of a comprehensive set of Smoothed Particle Hydrodynamics simulations of close encounters of white dwarfs aimed to explore the outcome of the interaction and the resulting gravitational wave patterns for different initial conditions. Depending on the initial conditions and the white dwarf masses, three different outcomes are possible. Specifically, the outcome of the interaction can be either a direct or a lateral collision or the interaction can result in the formation of an eccentric binary system. In those cases in which a collision occurs, the infalling material is compressed and heated and the physical conditions for a detonation are reached during the most violent phases of the merger in a significant number of simulations, whereas in some other, although some nuclear processing occurs in the region where the material of the disrupted less massive star hits the surface of the more massive one, the temperature increase in the shocked region rapidly lifts degeneracy, and the burning is quenched. We thus characterize under which circumstances a detonation is likely to occur as a result of the impact of the disrupted star on the surface of the more massive white dwarf. Finally, we also study which interactions result in bound systems, and in which ones the more massive white dwarf is also disrupted as a consequence of the dynamical interaction. In all cases we also compute the resulting gravitational wave emission.