  Covariant scalar fields exhibit divergences when quantized in two or more
spacetime dimensions: n \geg 2. Does perturbation theory, effective theories,
the renormalization group, etc., tell us all there is to know about these
problems? An alternative approach identifies the cause of divergences as due to
the effort to multiplicatively relate two measures that are mutually singular,
while the cure for those divergences is to introduce an O(\hbar) counterterm
that converts mutually singular measures into equivalent measures. This
procedure leads to a nontrivial, divergence-free formulation for all n \geq 2.
Finally, a critical comparison of the new methods with traditional procedures
is given.
