  Perturbative renormalization group theory is developed as a unified tool for
global asymptotic analysis. With numerous examples, we illustrate its
application to ordinary differential equation problems involving multiple
scales, boundary layers with technically difficult asymptotic matching, and WKB
analysis. In contrast to conventional methods, the renormalization group
approach requires neither {\it ad hoc\/} assumptions about the structure of
perturbation series nor the use of asymptotic matching. Our renormalization
group approach provides approximate solutions which are practically superior to
those obtained conventionally, although the latter can be reproduced, if
desired, by appropriate expansion of the renormalization group approximant. We
show that the renormalization group equation may be interpreted as an amplitude
equation, and from this point of view develop reductive perturbation theory for
partial differential equations describing spatially-extended systems near
bifurcation points, deriving both amplitude equations and the center manifold.
