This volume is based upon work done by the author and his collaborators over a period of approximately twenty years and is dedicated to some selected topics of quantum field theory, which have proved of increasing importance with the passing of time. There are three parts: Combinatoric Methods; Equations for Green's Functions and Perturbative Expansions; Regularization, Renormalization, and Mass Equations. This work will be useful to anyone interested in learning or using quantum field theory, many-body physics and, as well, to many applied mathematicians, because it introduces a number of combinatoric and analytic tools which drastically simplify, and at times bypass, treatments which customarily take up most of the bulk of the standard texts. It is part of the Frontiers in Physics series, edited by David Pines.