  Covariant, self-interacting scalar quantum field theories admit solutions for
low enough spacetime dimensions, but when additional divergences appear in
higher dimensions, the traditional approach leads to results, such as
triviality, that are less than satisfactory. Guided by idealized but soluble
{\it non}renormalizable models, a nontraditional proposal for the quantization
of covariant scalar field theories is advanced, which achieves a term-by-term,
divergence-free, perturbation analysis of interacting models expanded about a
suitable pseudofree theory, which differs from a free theory by an O(\hbar^2)
counterterm. These positive features are realized within a functional integral
formulation by a local, nonclassical, counterterm that effectively transforms
parameter changes in the action from generating mutually singular measures,
which are the basis for divergences, to equivalent measures, thereby removing
all divergences. The use of an alternative model about which to perturb is
already supported by properties of the classical theory, and is allowed by the
inherent ambiguity in the quantization process itself. This procedure not only
provides acceptable solutions for models for which no acceptable, faithful
solution currently exists, e.g., \phi^4_n, for spacetime dimensions n\ge4, but
offers a new, divergence-free solution, for less-singular models as well, e.g.,
\phi^4_n, for n=2,3. Our analysis implies similar properties for multicomponent
scalar models, such as those associated with the Higgs model.
