  Conventional wisdom holds that no four-dimensional gravity field theory can
be ultraviolet finite. This understanding is based mainly on power counting.
Recent studies confirm that one-loop N = 8 supergravity amplitudes satisfy the
so-called `no-triangle hypothesis', which states that triangle and bubble
integrals cancel from these amplitudes. A consequence of this hypothesis is
that for any number of external legs, at one loop N = 8 supergravity and N = 4
super-Yang-Mills have identical superficial degrees of ultraviolet behavior in
D dimensions. We describe how the unitarity method allows us to promote these
one-loop cancellations to higher loops, suggesting that previous power counts
were too conservative. We discuss higher-loop evidence suggesting that N = 8
supergravity has the same degree of divergence as N = 4 super-Yang-Mills theory
and is ultraviolet finite in four dimensions. We comment on calculations needed
to reinforce this proposal, which are feasible using the unitarity method.
