  The phase transition to superfluidity and the BCS-BEC crossover for an
ultracold gas of fermionic atoms is discussed within a functional
renormalization group approach. Non-perturbative flow equations, based on an
exact renormalization group equation, describe the scale dependence of the
flowing or average action. They interpolate continuously from the microphysics
at atomic or molecular distance scales to the macroscopic physics at much
larger length scales, as given by the interparticle distance, the correlation
length, or the size of the experimental probe. We discuss the phase diagram as
a function of the scattering length and the temperature and compute the gap,
the correlation length and the scattering length for molecules. Close to the
critical temperature, we find the expected universal behavior. Our approach
allows for a description of the few-body physics (scattering and molecular
binding) and the many-body physics within the same formalism.
