The chapter discusses the family of bounded analytic functions \( B(D) \) in a domain \( D \) of the complex plane, where \( f(z) \) is analytic and single-valued in \( D \) with \( |f(z)| < 1 \) for all \( z \in D \). Classical investigations were primarily focused on simply-connected domains, often the unit disk, due to the simplicity of its properties. However, the monodromy theorem fails in multiply-connected domains, leading to new challenges. The chapter explores the extension of classical results to multiply-connected domains, including the work of Carlson, Teichmüller, and Heins. Grunsky's contributions are highlighted, particularly his generalization of the Schwarz lemma for functions in multiply-connected domains. Ahlfors further developed these ideas, providing a characterization of extremal mappings and their properties. The extremal problems for functions in \( B(D) \) are connected to dual minimum problems, and the extremal functions yield mappings of \( D \) onto covering surfaces of the unit circle. The chapter also discusses the classes \( P \) and \( B_R \), which are related to \( B(D) \) through transformations, and their applications in solving extremal problems. Generalizations of \( B(D) \) are introduced, such as functions with a bounded real part or complex harmonic functions, and methods for solving extremal problems in these classes are outlined. The chapter concludes with references to key works in the field, emphasizing the significance of these contributions in the theory of bounded analytic functions.The chapter discusses the family of bounded analytic functions \( B(D) \) in a domain \( D \) of the complex plane, where \( f(z) \) is analytic and single-valued in \( D \) with \( |f(z)| < 1 \) for all \( z \in D \). Classical investigations were primarily focused on simply-connected domains, often the unit disk, due to the simplicity of its properties. However, the monodromy theorem fails in multiply-connected domains, leading to new challenges. The chapter explores the extension of classical results to multiply-connected domains, including the work of Carlson, Teichmüller, and Heins. Grunsky's contributions are highlighted, particularly his generalization of the Schwarz lemma for functions in multiply-connected domains. Ahlfors further developed these ideas, providing a characterization of extremal mappings and their properties. The extremal problems for functions in \( B(D) \) are connected to dual minimum problems, and the extremal functions yield mappings of \( D \) onto covering surfaces of the unit circle. The chapter also discusses the classes \( P \) and \( B_R \), which are related to \( B(D) \) through transformations, and their applications in solving extremal problems. Generalizations of \( B(D) \) are introduced, such as functions with a bounded real part or complex harmonic functions, and methods for solving extremal problems in these classes are outlined. The chapter concludes with references to key works in the field, emphasizing the significance of these contributions in the theory of bounded analytic functions.