  We define the renormalization group flow for a renormalizable interacting
quantum field in curved spacetime via its behavior under scaling of the
spacetime metric, $\g \to \lambda^2 \g$. We consider explicitly the case of a
scalar field, $\phi$, with a self-interaction of the form $\kappa \phi^4$,
although our results should generalize straightforwardly to other
renormalizable theories. We construct the interacting field--as well as its
Wick powers and their time-ordered-products--as formal power series in the
algebra generated by the Wick powers and time-ordered-products of the free
field, and we determine the changes in the interacting field observables
resulting from changes in the renormalization prescription. Our main result is
the proof that, for any fixed renormalization prescription, the interacting
field algebra for the spacetime $(M, \lambda^2 \g)$ with coupling parameters
$p$ is isomorphic to the interacting field algebra for the spacetime $(M, \g)$
but with different values, $p(\lambda)$, of the coupling parameters. The map $p
\to p(\lambda)$ yields the renormalization group flow. The notion of essential
and inessential coupling parameters is defined, and we define the notion of a
fixed point as a point, $p$, in the parameter space for which there is no
change in essential parameters under renormalization group flow.
