08-11U

An overview of the main approaches to study the $2D$ $XY$ model properties, especially the vortex influence, is presented. The well known results for the pure model are combined with the original results for a model with frozen site-dilution obtained by the author with collaborators. Namely, the overview covers in a brief form the following topics: the spin-wave approximation for the $2D$ $XY$ model, the basic principles of the phenomenological theory of topological defects, the Villain model. In the conclusion a simple estimation of the critical temperature is done to obtain the diluted model phase diagram.