Abstract: We consider several models of particles with short-range attractive interactions whose universal properties are controlled by an unstable renormalization-group fixed point at zero density and temperature. The fixed point corresponds to the Feshbach resonance, and relevant perturbations are the detuning of the resonance, and parameters that control the particle densities. Some critical exponents are determined exactly, and scaling functions are expressed as expansions about two and four spatial dimensions. The existence of a renormalization-group fixed point implies a universal phase diagram as a function of density, temperature, population imbalance, and detuning. We study this phase diagram in the context of BEC-BCS crossover of s-wave paired fermions. We develop a 1/N expansion, based upon models with Sp(2N) symmetry, which overcomes certain limitations of the expansions about two and four dimensions. Using this approach we chart the phase diagram of uniform ultra-cold fermions with population imbalance, as well as the superfluid-insulator phase diagram of balanced fermions in a three-dimensional optical lattice.