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This book develops the theory of stochastic relations as a basis for Markov transition systems. After an introduction to the basic mathematical tools from topology, measure theory, and categories, it examines congruences and morphisms. It applies these topics to the monoidal structure, and defines bisimilarity and behavioral equivalence within this framework. Developments from the general theory of coalgebras in the context of the subprobability functor are presented. The book also includes case studies of software architecture, the converse of a stochastic relation, and the average case analysis of two algorithms.