This book is devoted to the global pseudo-differential calculus on Euclidean spaces and its applications to non-commutative 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 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. The second main line is the self-contained presentation, for the first time in 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. --Book Jacket.