Pseudo Differential Operators and Symmetries : Background Analysis and Advanced Topics This monograph is devoted to the development of the theory of pseudo dierential n operators on spaces with symmetries. Such spaces are the Euclidean space R the n torus T compact Lie groups and compact homogeneous spaces. The book consists of several parts. One of our aims has been not only to present new results on pseudo dierential operators but also to show parallels between dierent approaches to pseudo dierential operators on dierent spaces. Moreover we tried to present the material in a self contained way to make it accessible for readers approaching the material for the rst time. However dierent spaces on which we develop the theory of pseudo dier tial operators require dierent backgrounds. Thus while operators on the clidean space in Chapter 2 rely on the well known Euclidean Fourier analysis pseudo dierentialoperatorsonthetorusandmoregeneralLiegroupsinChapters 4 and 10 require certain backgrounds in discrete analysis and in the representation theory of compact Lie groups which we therefore present in Chapter 3 and in Part III respectively. Moreover anyonewhowishestoworkwithpseudo dierential erators on Lie groups will certainly benet from a good grasp of certain aspects of representation theory. That is why we present the main elements of this theory in Part III thus eliminating the necessity for the reader to consult other sources for most of the time. Similarly the backgrounds for the theory of pseudo dierential 3 operators on S and SU(2) developed in Chapter 12 can be found in Chapter 11 presented in a self contained way suitable for immediate use. Foundations of Analysis Sets Topology and Metrics Elementary Functional Analysis Measure Theory and Integration Algebras Commutative Symmetries Fourier Analysis on n Pseudo differential Operators on n Periodic and Discrete Analysis Pseudo differential Operators on Commutator Characterisation of Pseudo differential Operators Representation Theory of Compact Groups Groups Topological Groups Linear Lie Groups Hopf Algebras Non commutative Symmetries Pseudo differential Operators on Compact Lie Groups Fourier Analysis on SU(2) Pseudo differential Operators on SU(2) Pseudo differential Operators on Homogeneous Spaces.