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

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... (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.