  The physics of atomic quantum gases is currently taking advantage of a
powerful tool, the possibility to fully adjust the interaction strength between
atoms using a magnetically controlled Feshbach resonance. For fermions with two
internal states, formally two opposite spin states, this allows to prepare long
lived strongly interacting three-dimensional gases and to study the BEC-BCS
crossover. Of particular interest along the BEC-BCS crossover is the so-called
unitary gas, where the atomic interaction potential between the opposite spin
states has virtually an infinite scattering length and a zero range. This
unitary gas is the main subject of the present chapter: It has fascinating
symmetry properties, from a simple scaling invariance, to a more subtle
dynamical symmetry in an isotropic harmonic trap, which is linked to a
separability of the N-body problem in hyperspherical coordinates. Other
analytical results, valid over the whole BEC-BCS crossover, are presented,
establishing a connection between three recently measured quantities, the tail
of the momentum distribution, the short range part of the pair distribution
function and the mean number of closed channel molecules.
