Global Pseudo Differential Calculus on Euclidean Spaces Background meterial Global Pseudo Differential Calculus  Pseudo Differential Operators and H Polynomials G Pseudo Differential Operators Spectral Theory Non Commutative Residue and Dixmier Trace Exponential Decay and Holomorphic Extension of Solutions. This book is devoted to the global pseudo differential calculus on Euclidean spaces and its applications to geometry and mathematical physics with emphasis on operators of linear and non linear quantum physics and travelling waves equations. The pseudo differential calculus presented here has an elementary character being addressed to a large audience of scientists. It includes the standard classes with global homogeneous structures the so called G and gamma operators. Concerning results for the applications a first main line is represented by spectral theory. Beside complex powers of operators and asymptotics for the counting function particular attention is here devoted to the non commutative residue in Euclidean spaces and the Dixmier trace. Second main line is the self contained presentation for the first time in a text book form of the problem of the holomorphic extension of the solutions of the semi linear globally elliptic equations. Entire extensions are discussed in detail. Exponential decay is simultaneously studied.