  The (mean field based) BCS theory is considered one of the most successful
theories in condensed matter physics. It is justified in ordinary metal
superconductors the coherence length $\xi$ is large, with two important
features: the order parameter (OP) and excitation gap (EG) are identical, and
the pair formation and their Bose condensation take place at the same
temperature Tc. It fails to explain the underdoped cuprate superconductivity:
EG is finite at Tc and thus distinct from OP. Since these superconductors
belong to a large class of small $\xi$ materials, this failure has the
potential for widespread impact.
  Here we have extended BCS theory in a natural way to short $\xi$
superconductors, based on a BCS--BEC crossover scenario, and arrived at a
simple physical picture in which incoherent, finite momentum pairs become
progressively more important as the pairing interaction becomes stronger,
leading to the distinction between EG and OP. The superconductivity from the
fermionic perspective and BEC from the bosonic perspective are just two sides
of the same coin.
  Our theory is capable of making verifiable quantitative predictions. We
obtain a cuprate phase diagram (with one free parameter) , in
(semi-)quantitative agreement with experiment. The mutually compensating
contributions from fermionic quasiparticles and bosonic pair excitations
provides a natural explanation for the quasi-universal behavior of the in-plane
superfluid density versus T. Our bosonic pair excitations also provide an
intrinsic mechanism for the long mysterious linear T terms in the specific
heat. Incoherent pair contributions lead to new low T power laws, consistent
with existing experiments. Finally, we demonstrated that the onset of
superconducting long range order leads to sharp features in the specific heat
at Tc, consistent with experiment.
