  It has long been known that particles with short-range repulsive interactions
in spatial dimension d=1 form universal quantum liquids in the low density
limit: all properties can be related to those of the spinless free Fermi gas.
Previous renormalization group (RG) analyses demonstrated that this
universality is described by an RG fixed point, infrared stable for d<2, of the
zero density gas. We show that for d>2 the same fixed point describes the
universal properties of particles with short-range attractive interactions near
a Feshbach resonance; the fixed point is now infrared unstable, and the
relevant perturbation is the detuning of the resonance. Some exponents are
determined exactly, and the same expansion in powers of (d-2) applies for
scaling functions for d<2 and d>2. A separate exact RG analysis of a field
theory of the particles coupled to `molecules' finds an alternative description
of the same fixed point, with identical exponents; this approach yields a (4-d)
expansion which agrees with the recent results of Nishida and Son
(cond-mat/0604500). The existence of the RG fixed point implies a universal
phase diagram as a function of density, temperature, population imbalance, and
detuning; in particular, this applies to the BEC-BCS crossover of fermions with
s-wave pairing. Our results open the way towards computation of these universal
properties using the standard field-theoretic techniques of critical phenomena,
along with a systematic analysis of corrections to universality. We also
propose a 1/N expansion (based upon models with Sp(2N) symmetry) of the fixed
point and its vicinity, and use it to obtain results for the phase diagram.
