  This chapter is a pedagogical review of the Hubbard model for bosons with
repulsion and for fermions with attraction and repulsion primarily using two
methods, one chosen for its simplicity and insights (mean field theory) and the
other chosen for its accuracy and reliability (quantum Monte Marlo methods).
From a comparison of the two methods we glean valuable information into the
effects of fluctuations that dominate quantum phase transitions. The chapter
includes an in-depth comparison with experiments. We conclude with a discussion
of future developments where the technical methods expounded on here, mean
field theory and quantum Monte Carlo, could be useful.
