This monograph is devoted to topical issues of modern mapping theory, and totals four chapters. The first three are devoted only to issues of mapping theory, while the last concerns mapping theory and applications to differential equations. All chapters are in one way or another connected with the modulus technique and the possibility of its application to mappings. The problems considered in this monograph are classical issues of analysis: distance estimates under mappings, the possibility of continuous boundary extension, equicontinuity of families, the behavior of mappings in metric spaces, etc. The book can be read starting from any section and any chapter, since chapters are not directly connected with each other, for the convenience of the reader.

The book has a total of 498 pages. It is written at a high scholarly level and in accessible language. It is a continuation of the research presented in the author's previous monograph: Mappings with Direct and Inverse Poletsky Inequalities.

This monograph was supported by the National Research Foundation of Ukraine (Project "Analogues of Caratheodory and Koebe-Bloch theorems for Orlycz-Sobolev classes," Project number 2025.02/0010).

Evgeny Sevost'yanov is Head of Department of Mathematical Analysis, Business Analysis and Statistics at Zhytomyr Ivan Franko State University, Ukraine. His research interests include mapping theory and its applications to equations with partitional derivatives. The main results of E. Sevost'yanov concern the local and boundary behavior of mappings, mappings of metric spaces and Riemannian manifolds, quotient spaces and Riemannian surfaces. From 2002 to 2022, E. Sevost'yanov worked at the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine as a researcher," please change 2022 to 2014. That is, it should read "From 2002 to 2014, E. Sevost'yanov worked at the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine as a researcher.