  We construct systematic expansions around four and two spatial dimensions for
a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi
gas can be understood as a weakly interacting system of fermionic and bosonic
degrees of freedom. To the leading and next-to-leading orders in the expansion
over epsilon=4-d, with d being the dimensionality of space, we calculate the
thermodynamic quantities and the fermion quasiparticle spectrum, both at
unitarity and around the unitarity point. Then the phase structure of the
polarized Fermi gas in the unitary regime is studied in the epsilon expansion.
At unitarity the unpolarized superfluid state and a fully polarized normal
state are separated by a first order phase transition. However, on the BEC side
of the unitarity point, in a certain range of the two-body binding energy,
these two phases are separated in the phase diagram by more exotic phases,
including gapless superfluid phases with one or two Fermi surfaces and a
superfluid phase with a spatially varying condensate. We also show that the
unitary Fermi gas near two spatial dimensions is weakly interacting, and
calculate the thermodynamic quantities and the fermion quasiparticle spectrum
in the expansion over \bar\epsilon=d-2.
