This book presents a comprehensive treatment of one-parameter semigroups of positive linear operators on ordered Banach spaces. It is a collection of contributions by leading experts in the field, edited by Rainer Nagel. The work is organized into four parts, each focusing on different types of underlying spaces: Banach spaces, spaces C₀(X), Banach lattices, and non-commutative operator algebras such as C*- and W*-algebras. The book covers three main topics: characterization of semigroups, spectral theory, and asymptotic behavior. The first part, on Banach spaces, includes basic results, characterization of semigroups, spectral theory, and asymptotic behavior. The second part focuses on spaces C₀(X), covering similar topics. The third part deals with Banach lattices, while the fourth part addresses positive semigroups on C*- and W*-algebras. Each section begins with a summary of basic results and notations, allowing readers to navigate the content according to their interests. The book also includes a table of contents, a bibliography, and a table of symbols. The authors emphasize the importance of connecting the theory of positive operators with the study of Cauchy problems with positive solutions. The work is intended to provide a coherent theory of one-parameter semigroups of positive linear operators on ordered Banach spaces, with applications in areas such as probability theory, partial differential equations, and mathematical physics.This book presents a comprehensive treatment of one-parameter semigroups of positive linear operators on ordered Banach spaces. It is a collection of contributions by leading experts in the field, edited by Rainer Nagel. The work is organized into four parts, each focusing on different types of underlying spaces: Banach spaces, spaces C₀(X), Banach lattices, and non-commutative operator algebras such as C*- and W*-algebras. The book covers three main topics: characterization of semigroups, spectral theory, and asymptotic behavior. The first part, on Banach spaces, includes basic results, characterization of semigroups, spectral theory, and asymptotic behavior. The second part focuses on spaces C₀(X), covering similar topics. The third part deals with Banach lattices, while the fourth part addresses positive semigroups on C*- and W*-algebras. Each section begins with a summary of basic results and notations, allowing readers to navigate the content according to their interests. The book also includes a table of contents, a bibliography, and a table of symbols. The authors emphasize the importance of connecting the theory of positive operators with the study of Cauchy problems with positive solutions. The work is intended to provide a coherent theory of one-parameter semigroups of positive linear operators on ordered Banach spaces, with applications in areas such as probability theory, partial differential equations, and mathematical physics.