Presenting some recent results on the construction and the moments of Levy-type processes, the focus of this volume is on a new existence theorem, which is proved using a parametrix construction. Applications range from heat kernel estimates for a class of Levy-type processes to existence and uniqueness theorems for Levy-driven stochastic differential equations with Hoelder continuous coefficients. Moreover, necessary and sufficient conditions for the existence of moments of Levy-type processes are studied and some estimates on moments are derived. Levy-type processes behave locally like Levy processes but, in contrast to Levy processes, they are not homogeneous in space. Typical examples are processes with varying index of stability and solutions of Levy-driven stochastic differential equations. This is the sixth volume in a subseries of the Lecture Notes in Mathematics called Levy Matters. Each volume describes a number of important topics in the theory or applications of Levy processes and pays tribute to the state of the art of this rapidly evolving subject, with special emphasis on the non-Brownian world.