Complex analytic spaces and more generally Thom-Mather stratified spaces provide an interesting class of singular spaces with a rich interaction of analysis, geometry and topology. The aim of this talk is to describe my research activity which has been mainly concerned with the spectral theory and L2-cohomology of such spaces. After a brief introduction to the subject I will explain some results concerning the existence of self-adjoint extensions with entirely discrete spectrum of the Hodge and the Dolbeault Laplacian on compact complex spaces. Then I will describe some topological applications of these results. Finally in the last part I will explain a result concerning the existence of an L2-de Rham complex with Poincare' duality on Thom-Mather stratified spaces.