  We derive a theory of superfluidity for a dilute Fermi gas that is valid when
scattering resonances are present. The treatment of a resonance in many-body
atomic physics requires a novel mean-field approach starting from an
unconventional microscopic Hamiltonian. The mean-field equations incorporate
the microscopic scattering physics, and the solutions to these equations
reproduce the energy-dependent scattering properties. This theory describes the
high-$T_c$ behavior of the system, and predicts a value of $T_c$ which is a
significant fraction of the Fermi temperature. It is shown that this novel
mean-field approach does not break down for typical experimental circumstances,
even at detunings close to resonance. As an example of the application of our
theory we investigate the feasibility for achieving superfluidity in an
ultracold gas of fermionic $^6$Li.
