Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We briefly describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topological models, a spacial case of neighborhood semantics. A new cladding for bisimulation is presented. The concept of Alexandroff peritopology is used in order to determine the logic of all peritopological spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T-0, T-1 and T-2-peritopological spaces are not modal definable, and that D is the logic of all proper peritopological spaces. Finally, among our conclusions, we show that the question whether T-0, T-1 peritopological spaces are modal definable in H(@) remains open.