Derived categories and K-groups of singular varieties

This thesis consists of three parts and is a collection of papers written by the author of this text during his postgraduate studies, together with an Appendix chapter.
The first chapter is based on [98] and is in collaboration with Evgeny Shinder. It discusses the K-groups K_1, K_0 and K_{−n} of the singularity category of isolated quotient singularities. The second chapter is based on [73] and is joint with Martin Kalck and Evgeny Shinder. It introduces Kawamata type semiorthogonal decompositions for singular varieties and obstructions for such decompositions are studied, mainly for the case of nodal threefolds. Each of these two chapters can be read independently. The third chapter is an Appendix to the first chapter and explains in more detail how the main technical result in chapter one is proven, on which the main theorems rely on.