Abstract: In the course of development of viscous fluid dynamics, the simplified theories of the Navier-Stokes (NS) equations, corresponding algorithms and applications form the frontier and the core of viscous fluid dynamics in different historical period. With that point in mind we review here some advances in the classical boundary layer, multilayer boundary layer (triple deck), interacting boundary layer and diffusion parabolized NS equations theories and corresponding algorithms and applications. The relationship between those theories and experiments, a note to simplifying calculation of turbulent flow and some problems of combining numerical simulation with physical analysis are also reviewed.