  On the basis of the observed short coherence lengths in the cuprates we argue
that a BCS-Bose-Einstein condensation (BEC) crossover approach is an
appropriate starting point for correcting the mean field approach of BCS and,
thereby, for addressing pseudogap phenomena in these materials. Our version of
the BCS-BEC approach is based on a particular Greens' function decoupling
scheme which should be differentiated from others in the literature, and which
yields (i) the Leggett crossover ground state (for all coupling constants g, at
T=0) and (ii) BCS theory (for all $ T \leq T_c$ over a range of small g). In
this paper we provide a simple physical picture of the pseudogap phase above
and below $T_c$, and review the quantitative and qualitative implications of
this theory, which, for the most part have been published in a series of recent
papers.
