This work covers various topics in nonlinear boundary value problems for holomorphic functions, including existence and uniqueness, results, questions concerning parameter dependence, regularity theorems, several procedures for numerically solving such problems, and applications to nonlinear singular integral equations. The emphasis is mainly on the geometric aspects of the matter. A key role is played by an appropriate generalization of the classical maximum principle, which establishes intimate connections between boundary value problems and extremal problems and also opens a novel approach to interpolation and approximation with holomorphic functions and to Ho-optimization. In the investigation of nonlinear singular integral equations the interest focuses on the structure of the solution manifold. Bifurcation phenomena are not merely detected, but also classified in the framework of singularity theory. Numerical methods for solving the problems considered are proposed and some experience gained in their realization is reported. The book requires only elementary knowledge of function theory, the necessary preliminaries are summarized in a separate chapter.