This book presents the technique of pseudodifferential operators and its applications, especially to the Dirac theory of quantum mechanics. The treatment uses ''Leibniz' formulas'' with integral remainders or as asymptotic series. A pseudodifferential operator may also be described by invariance under action of a Lie-group. The author discusses connections to the theory of C*-algebras, invariant algebras of pseudodifferential operators under hyperbolic evolution, and the relation of the hyperbolic theory to the propagation of maximal ideals.