Metrics on the Phase Space and Non Selfadjoint Pseudo Differential Operators Basic Notions of Phase Space Analysis Metrics on the Phase Space Estimates for Non Selfadjoint Operators. This book is devoted to the study of pseudo dierential operators with special emphasis on non selfadjoint operators a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi classical estimates for non selfadjoint operators. Therstchapter Basic Notions of Phase Space Analysis isintroductoryand gives a presentation of very classical classes of pseudo dierential operators along with some basic properties. As an illustration of the power of these methods we give a proof of propagation of singularities for real principal type operators (using aprioriestimates andnotFourierintegraloperators) andweintroducethereader to local solvability problems. That chapter should be useful for a reader say at the graduate level in analysis eager to learn some basics on pseudo dierential operators. The second chapter Metrics on the Phase Space begins with a review of symplectic algebra Wigner functions quantization formulas metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space following essentially the basic assumptions of L. H¨ ormander (Chapter 18 in the book [73]) on this topic.