Poisson Structures Part I Theoretical Background:1.Poisson Structures: Basic Definitions 2.Poisson Structures: Basic Constructions 3.Multi Derivations and Kähler Forms 4.Poisson (Co)Homology 5.Reduction Part II Examples:6.Constant Poisson Structures Regular and Symplectic Manifolds 7.Linear Poisson Structures and Lie Algebras 8.Higher Degree Poisson Structures 9.Poisson Structures in Dimensions Two and Three 10.R Brackets and r Brackets 11.Poisson–Lie Groups Part III Applications:12.Liouville Integrable Systems 13.Deformation Quantization A Multilinear Algebra B Real and Complex Differential Geometry References Index List of Notations. . Poisson structures appear in a large variety of contexts ranging from string theory classical quantum mechanics and differential geometry to abstract algebra algebraic geometry and representation theory. In each one of these contexts it turns out that the Poisson structure is not a theoretical artifact but a key element which unsolicited comes along with the problem that is investigated and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.