  We develop a functional integral formalism for ultracold gases of fermionic
atoms. It describes the BEC - BCS crossover and involves both atom and molecule
fields. Beyond mean field theory we include the fluctuations of the molecule
field by the solution of gap equations. In the BEC limit, we find that the low
temperature behavior is described by a Bogoliubov theory for bosons. For a
narrow Feshbach resonance these bosons can be associated with microscopic
molecules. In contrast, for a broad resonance the interaction between the atoms
is approximately pointlike and microscopic molecules are irrelevant. The bosons
represent now correlated atom pairs or composite ``dressed molecules''. The low
temperature results agree with quantum Monte Carlo simulations. Our formalism
can treat with general inhomogeneous situations in a trap. For not too strong
inhomogeneities the detailed properties of the trap are not needed for the
computation of the fluctuation effects - they enter only in the solutions of
the field equations.
