This book is concerned with the theory of model representations of linear non-selfadjoint and non-unitary operators. This booming area of functional analysis owes its origins to the fundamental works of M. S. Livšic on the theory of characteristic functions, the deep studies of B. S.-Nagy and C. Foias on dilation theory, and also to the Lax–Phillips scattering theory. Here, a uniform conceptual approach is developed which organically unites all these theories. New analytic methods are introduced which make it possible to solve some important problems from the theory of spectral representations. Aimed at specialists in functional analysis, the book will also be accessible to senior mathematics students.