In this survey, we study applications of combinatorial methods in the ring theory. Unlike the well-known survey by Ufnarovskii, we are more interested in the technique of obtaining these methods and not in the results themselves. Therefore, the theme of this survey is narrower and is more in correspondence with the authors' interests. The main subject here is the word combinatorial analysis and its applications, and also the notion of the canonical form of elements in different calculi.
An element in some algebra can be represented as a linear combination of words; therefore, the problems of the word combinatorial analysis are the most "pure." The study of the growth of algebras and Hilbert series can also be reduced to the word combinatorial analysis, also as periodicity effects, which appear in problems of the Burnside type (the Shirshov height theorem, independency theorem, Pi-algebra radical nilpotency, and so on).