  Both the trapping geometry and the interatomic interaction strength of a
dilute ultracold fermionic gas can be well controlled experimentally. When the
interactions are tuned to strong attraction, Cooper pairing of neutral atoms
takes place and a BCS superfluid is created. Alternatively, the presence of
Feshbach resonances in the interatomic scattering allows populating a molecular
(bound) state. These molecules are more tightly bound than the Cooper pairs and
can form a Bose-Einstein condensate (BEC). In this contribution, we describe
both the BCS and BEC regimes, and the crossover, from a functional integral
point of view. In this description, the properties of the superfluid (such as
vortices and Josephson tunneling) can be derived and followed as the system is
tuned from BCS the BEC. In particular, we present results for the critical
current of the superfluid through an optical lattice and link these results to
recent experiments with atomic bosons in optical lattices.
