The main topic treated by this book is a study of the generalizations of the Poincaré-Birkhoff-Witt theorem to quadratic algebras. The book starts with a review of various definitions and facts from homological algebra (including the bar construction and a quadratic duality for quadratic algebras). In chapter 2, the authors describe various definitions and properties of Koszul algebras and Koszul modules. A notion of an infinitesimal bialgebra of a Koszul algebra is introduced here. Natural operations on quadratic algebras and modules are studied in chapter 3 (including the Segre product and Veronese powers). The PBW algebras form a special subclass of Koszul algebras and their properties are studied in chapter 4. Nonhomogeneous quadratic algebras are studied in chapter 5. In particular, there is a discussion of an analogue of quadratic duality leading to the notion of curved DG-algebras. The Kozsul deformation principle and its consequences are discussed in chapter 6. The final chapter is devoted to a surprising connection to one-dependent discrete-time stochastic processes. The book offers a nice review of the field together with some new results.

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