  The subject of BCS - Bose Einstein condensation (BEC) crossover is
particularly exciting because of its realization in ultracold Fermi gases and
its possible relevance to high temperature superconductors. In the paper we
review that body of theoretical work on this subject which represents a natural
extension of the seminal papers by Leggett and by Nozieres and Schmitt-Rink
(NSR). The former addressed only the ground state, now known as the
"BCS-Leggett" wave-function and the key contributions of the latter pertain to
calculations of the superfluid transition temperature $T_c$. These two papers
have given rise to two main and, importantly, distinct, theoretical schools in
the BCS-BEC crossover literature. The first of these extends the BCS-Leggett
ground state to finite temperature and the second extends the NSR scheme away
from $T_c$ both in the superfluid and normal phases. It is now rather widely
accepted that these extensions of NSR produce a different ground state than
that first introduced by Leggett. Our analysis shows how the NSR-based approach
views the bosonic contributions more completely but it treats the fermions as
"quasi-free". By contrast, the BCS-Leggett based approach treats the fermionic
contributions more completely but it treats the bosons as "quasi-free". The NSR
based schemes approach the crossover between BCS and BEC by starting from the
BEC limit and the BCS-Leggett based scheme approaches this crossover by
starting from the BCS limit. Ultimately, one would like to combine these two
schemes. In this paper we review the strengths and weaknesses of both
approaches. To reach a full understanding, it is important in the future to
invest effort in investigating in more detail the T=0 aspects of NSR-based
theory and the $T \neq 0$ aspects of BCS-Leggett theory.
