  We show how to enlarge the covariance group of any classical field theory in
such a way that the resulting "covariantized" theory is 'essentially
equivalent' to the original. In particular, our technique will render any
classical field theory generally covariant, that is, the covariantized theory
will be spacetime diffeomorphism-covariant and free of absolute objects. Our
results thus generalize the well-known parametrization technique of Dirac and
Kucha\v{r}. Our constructions apply equally well to internal covariance groups,
in which context they produce natural derivations of both the Utiyama minimal
coupling and St\"uckelberg tricks.
