  In this article we discuss our ongoing program to extend the scope of
certain, well-developed microlocal methods for the asymptotic solution of
Schr\"{o}dinger's equation (for suitable `nonlinear oscillatory' quantum
mechanical systems) to the treatment of several physically significant,
interacting quantum field theories. Our main focus is on applying these
`Euclidean-signature semi-classical' methods to self-interacting (real) scalar
fields of renormalizable type in 2, 3 and 4 spacetime dimensions and to
Yang-Mills fields in 3 and 4 spacetime dimensions. A central argument in favor
of our program is that the asymptotic methods for Schr\"{o}dinger operators
developed in the microlocal literature are far superior, for the quantum
mechanical systems to which they naturally apply, to the conventional WKB
methods of the physics literature and that these methods can be modified, by
techniques drawn from the calculus of variations and the analysis of elliptic
boundary value problems, to apply to certain (bosonic) quantum field theories.
Unlike conventional (Rayleigh/ Schr\"{o}dinger) perturbation theory these
methods allow one to avoid the artificial decomposition of an interacting
system into an approximating `unperturbed' system and its perturbation and
instead to keep the nonlinearities (and, if present gauge invariances) of an
interacting system intact at every level of the analysis.
