  In this paper we present a very general theoretical framework for addressing
fermionic superfluids over the entire range of BCS to Bose Einstein
condensation (BEC) crossover in the presence of population imbalance or spin
polarization. Our emphasis is on providing a theory which reduces to the
standard zero temperature mean field theories in the literature, but
necessarily includes pairing fluctuation effects at non-zero temperature within
a consistent framework. Physically, these effects are associated with the
presence of pre-formed pairs (or a fermionic pseudogap) in the normal phase,
and pair excitations of the condensate, in the superfluid phase. We show how
this finite $T$ theory of fermionic pair condensates bears many similarities to
the condensation of point bosons. In the process we examine three different
types of condensate: the usual breached pair or Sarma phase and both the one
and two plane wave Larkin- Ovchinnikov, Fulde-Ferrell (LOFF) states. The last
of these has been discussed in the literature albeit only within a
Landau-Ginzburg formalism, generally valid near $T_c$. Here we show how to
arrive at the two plane wave LOFF state in the ground state as well as at
general temperature $T$.
