  We study resonantly-paired s-wave superfluidity in a degenerate gas of two
species (hyperfine states labeled by $\uparrow,\downarrow$) of fermionic atoms
when the numbers $N_{\uparrow}$ and $N_{\downarrow}$ of the two species are
{\it unequal}, i.e., the system is "polarized". We find that the continuous
crossover from the Bose-Einstein condensate (BEC) limit of tightly-bound
diatomic molecules to the Bardeen-Cooper-Schrieffer (BCS) limit of weakly
correlated Cooper pairs, studied extensively at equal populations, is
interrupted by a variety of distinct phenomena under an imposed population
difference $\Delta N \equiv N_\uparrow - N_\downarrow$. Our findings are
summarized by a `` polarization\rq\rq ($\Delta N$) versus Feshbach-resonance
detuning ($\delta$) zero-temperature phase diagram, which exhibits regions of
phase separation, a periodic FFLO superfluid, a polarized normal Fermi gas and
a polarized molecular superfluid consisting of a molecular condensate and a
fully polarized Fermi gas. We describe numerous experimental signatures of such
phases and the transitions between them, in particular focusing on their
spatial structure in the inhomogeneous environment of an atomic trap.
