  This thesis presents theoretical work in s- and p-wave resonantly paired
Fermi gases at zero temperature. In the BEC regime of the wide-resonance s-wave
BCS-BEC crossover, the chemical potential, speed of sound, condensate
depletion, and ground state energy are computed. The quantities are calculated
in a low density approximation. The bosonic scattering length is computed
diagrammatically to be approximately 0.60a with a the fermionic scattering
length, confirming a well-known result. The perturbative approach breaks down
in the intermediate BCS-BEC crossover regime which is not investigated. Quantum
corrections are computed in the low density expansion, and the mass-imbalanced
two-species Fermi gas is investigated. For identical fermions, it is discussed
how p-wave Feshbach resonances naturally fall into two classes of "weak" and
"strong", depending on the short-distance physics. It is shown how bound
fermionic trimers appear in the strongly-resonant superfluid. The appearance of
the trimer under the strengthening of the resonance is investigated, and the
lifetime of p-wave diatomic molecules due to collisional relaxation into
trimers is estimated.
