  This review is written at the time of the twentieth anniversary of the
discovery of high temperature superconductors, which, nearly coincides with the
important discovery of the superfluid phases of ultracold trapped fermionic
atoms. We show how these two subjects have much in common. Both have been
addressed from the perspective of the BCS-Bose Einstein condensation (BEC)
crossover scenario, which is designed to treat short coherence length
superfluids with transition temperatures which are "high", with respect to the
Fermi energy. A generalized mean field treatment of BCS-BEC crossover at
general temperatures $T$, based on the BCS-Leggett ground state, has met with
remarkable success in the fermionic atomic systems. Here we summarize this
success in the context of four different cold atom experiments, all of which
provide indications, direct or indirect, for the existence of a pseudogap. This
scenario also provides a physical picture of the pseudogap phase in the
underdoped cuprates which is a central focus of high $T_c$ research. We
summarize successful applications of BCS-BEC crossover to key experiments in
high $T_c$ systems including the phase diagram, specific heat, and vortex core
STM data, along with the Nernst effect, and exciting recent data on the
superfluid density in very underdoped samples,
