This article aims to review the recent achievements and the state of the art in the high Reynolds number asymptotic theory of steady separated flow past bluff bodies for a general reader specializing in fluid dynamics who is not necessarily familiar with modern asymptotic techniques. A short historical overview is given. The ideas of the mathematical methods used are briefly outlined. Then the general structure of the solution for a plane flow past a bluff body is described. The physical mechanisms of such a flow are discussed, and quantitative results are given and compared with numerical calculations. Existing extensions of the theory and the latest results for axisymmetric flows are described. In conclusion, the relationship between asymptotic theory and real turbulent flows is discussed. This review article contains 76 references.