Multidimensional bosonization

Bosonization of degenerate fermions yields insight both into Landau-Fermi l iquids, and into non-Fermi liquids. We begin our review with a pedagogical introduction to bosonization, emphasizing its applicability in spatial dime nsions greater than one. After a brief historical overview, we present the essentials of the method. Well known results of Landau theory are recovered , demonstrating that this new tool of many-body theory is robust. Limits of multidimensional bosonization are tested by considering several examples o f non-Fermi liquids, in particular the composite fermion theory of the half -filled Landau level. Nested Fermi surfaces present a different challenge, and these may be relevant in the cuprate superconductors. We conclude by di scussing the future of multidimensional bosonization.