Schur functions in complex analysis are scalar-valued functions which are holomorphic and bounded by one on the unit disk of the complex plane. Generalized Schur functions are operator-valued functions which are holomorphic on a subregion of the unit disk containing the origin such that certain associated kernels have a finite number of negative squares. The book under review presents a theory of generalized Schur functions from an abstract operator theory viewpoint but also with a system theory flavor. The main tool is the realization of generalized Schur functions as characteristic functions of coisometric, isometric and unitary operator colligations whose state spaces are reproducing-kernel Pontryagin spaces. In the case that the values of the Schur function $S(z)$ are continuous operators between two Pontryagin spaces $\germ{F}$ and $\germ{G}$ having the same negative index, the graphs of the operator colligations are obtained as closures of linear relations. For these constructions to be valid, the assumption on the underlying spaces $\germ{F}$ and $\germ{G}$ is necessary. In the more general situation when $\germ{F}$ and $\germ{G}$ are Kre\u{ı}n spaces, each $S(z)$ admits a $2\times 2$ matrix representation $(S_{ij}(z))_{i,j=1,2}$ relative to fixed fundamental decompositions $\germ{F} = \germ{F}_+ \oplus \germ{F}_-$ and $\germ{G} =\germ{G}_+ \oplus \germ{G}_-$; if $S_{22}(0)$ is invertible then a method involving the Potapov-Ginzburg transform gives the operator colligations. The use of the Potapov-Ginzburg transform has the limitation that $S_{22}(0)$ is required to be invertible. However, modifications can be made to replace $0$ by any other point $\alpha$ in the unit disk such that $S_{22}(\alpha)$ is invertible. Since $S_{22}(z)$ is shown to be invertible except at a finite number of points, the realization theory, in this form, is applicable to any generalized Schur function. The realization theory for generalized Schur functions provides a setting for the relationship between factorization and invariant subspaces, operator models, Kre\u{ı}n-Langer and de Branges factorizations, and some other topics. \par Chapter 1 supplies background material on Kre\u{ı}n and Pontryagin spaces, reproducing-kernel Pontryagin spaces, operator colligations, Julia operators and contraction operators, linear relations, de Branges theory of complementary spaces. A key result that gives conditions for the closure of a linear relation to be the graph of an operator is also included. In Chapter 2 the authors introduce the generalized Schur functions after treating the realization problem for $\germ{L}(\germ{F},\germ{G})$-valued Schur functions in the case that $\germ{F}$ and $\germ{G}$ are Pontryagin spaces having the same negative index. Chapter 3 details the properties of the state spaces and gives examples and miscellaneous applications. In Chapter 4 the authors explore the relationship between factorization and invariant subspaces and construct realizations for arbitrary Schur functions by means of the Potapov-Ginzburg transform. Canonical models are also discussed. \par The book includes an epilogue with directions for further work, an appendix, notes that credit sources and point to additional literature, and an extensive list of references.