An overview on selected thermodynamic and kinetic aspects of the theoretical description of first-order phase transitions, and, in particular, of the theory of nucleation is given. Hereby both basic concepts as well as modern developments in this field are discussed. Details can be traced in the literature cited and in the given there further references. The analysis is directed here mainly to phase formation processes in non-crystalline materials. However, the majority of the methods employed and results discussed are, possibly with quantitative modifications, applicable to phase formation processes in any other fields of interest as well.