The book offered to the reader contains mathematically rigorous results regarding the construction and properties of calibration fields, both lattice and continuous. Many significant results belong here to the author himself. The book consists of two parts. The first part sets out the theory related to phase transitions in lattice models, the second - the theory of continuous gauge fields, more precisely, that part of it that is currently available to rigorous analysis. The supplement written by A.A. Migdal contains a number of unresolved problems that are important from the point of view of particle physics, and indicates possible approaches to their study.