  We present a theory of a degenerate atomic Fermi gas, interacting through a
narrow Feshbach resonance, whose position and therefore strength can be tuned
experimentally, as demonstrated recently in ultracold trapped atomic gases. The
distinguishing feature of the theory is that its accuracy is controlled by a
dimensionless parameter proportional to the ratio of the width of the resonance
to Fermi energy. The theory is therefore quantitatively accurate for a narrow
Feshbach resonance. In the case of a narrow s-wave resonance, our analysis
leads to a quantitative description of the crossover between a weakly-paired
BCS superconductor of overlapping Cooper pairs and a strongly-paired molecular
Bose-Einstein condensate of diatomic molecules. In the case of pairing via a
p-wave resonance, that we show is always narrow for a sufficiently low density,
we predict a detuning-temperature phase diagram, that in the course of a
BCS-BEC crossover can exhibit a host of thermodynamically-distinct phases
separated by quantum and classical phase transitions. For an intermediate
strength of the dipolar anisotropy, the system exhibits a p_x + i p_y paired
superfluidity that undergoes a topological phase transition between a
weakly-coupled gapless ground state at large positive detuning and a
strongly-paired fully-gapped molecular superfluid for a negative detuning. In
two dimensions the former state is characterized by a Pfaffian ground state
exhibiting topological order and non-Abelian vortex excitations familiar from
fractional quantum Hall systems.
