  As applied to quantum theories, the program of renormalization is successful
for `renormalizable models' but fails for `nonrenormalizable models'. After
some conceptual discussion and analysis, an enhanced program of renormalization
is proposed that is designed to bring the `nonrenormalizable models' under
control as well. The new principles are developed by studying several,
carefully chosen, soluble examples, and include a recognition of a `hard-core'
behavior of the interaction and, in special cases, an extremely elementary
procedure to remove the source of all divergences. Our discussion provides the
background for a recent proposal for a nontrivial quantization of
nonrenormalizable scalar quantum field models, which is briefly summarized as
well.
