Partial Differential Equations and Spectral Theory : [The idea for this volume originated from an International Conference on Partial Differential Equations and Spectral Theoryʺ held in Goslar Germany August 31 to September 06 2008] This volume collects six articles on selected topics at the frontier between partial differential equations and spectral theory written by leading specialists in their respective field. The articles focus on topics that are in the center of attention of current research with original contributions from the authors. They are written in a clear expository style that makes them accessible to a broader audience. The articles contain a detailed introduction and discuss recent progress provide additional motivation and develop the necessary tools. Moreover the authors share their views on future developments hypotheses and unsolved problems.