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This is an introduction to stratification theory, leading the reader up to modern research topics in the field. The first part presents the basics of stratification theory. In the second part the methods established in the first part are applied to particular classes of stratified spaces.

This is the first exposition of the quantization theory of singular symplectic (Marsden-Weinstein) quotients and their applications to physics. The reader will acquire an introduction to the various techniques used in this area, as well as an overview of the latest research approaches. These involve classical differential and algebraic geometry, as well as operator algebras and noncommutative geometry. Thus one will be amply prepared to follow future developments in this field.