the classical koszul theory plays an important role in the representation theory of graded algebras. however, there are a lot of structures (algebras, categories, etc) having natural gradings with non-semisimple degree 0 parts, to which the classical theory cannot apply. particular examples include polynomial rings over non-semisimple algebras, extension algebras of modules, etc. in this talk i'll introduce a generalized koszul theory which does not demand the semisimple property. it preserves many classical results as koszul duality and has a close relation to the classical one. applications of this generalized theory to extension algebras of modules and modular skew group algebras will be described.