Quasiregular Mappings extend quasiconformal theory to thenoninjective case.They give a natural and beautifulgeneralization of the geometric aspects ofthe theory ofanalytic functions of one complex variable to Euclideann-space or, more generally, to Riemannian n-manifolds. Thisbook is a self-contained exposition of the subject. A braodspectrum of results of both analytic and geometric characterare presented, and the methods vary accordingly. The maintools are the variational integral method and the extremallength method, both of which are thoroughly developed here.Reshetnyak's basic theorem on discreteness and openness isused from the beginning, but the proof by means ofvariational integrals is postponed until near the end. Thus,the method of extremal length is being used at an earlystage and leads, among other things, to geometric proofs ofPicard-type theorems and a defect relation, which are someof the high points of the present book.