Modern enumeration theory of permutations with restricted positions

This article reviews the results obtained by native and foreign authors for the most part in the last decade, these results are interpreted in the context of both classical and new trends of the enumeration theory of permutations with restricted positions which is a section of combinatorial analysis founded by Euler. The material of the review is given in accordance with the author's classification of the main trends of the theory. A series of previously announced results of the author, which generalize the well-known publications of Western mathematicians, are presented in detail.