  (Revised, with postscript figures appended, corrections and added comments.)
We develop and describe new approaches to the problem of interacting Fermions
in spatial dimensions greater than one. These approaches are based on
generalizations of powerful tools previously applied to problems in one spatial
dimension. We begin with a review of one-dimensional interacting Fermions. We
then introduce a simplified model in two spatial dimensions to study the role
that spin and perfect nesting play in destabilizing Fermion liquids. The
complicated functional renormalization group equations of the full problem are
made tractable in our model by replacing the continuum of points that make up
the closed Fermi line with four Fermi points. Despite this drastic
approximation, the model exhibits physically reasonable behavior both at
half-filling (where instabilities occur) and away from half-filling (where a
Luttinger liquid arises). Next we implement the Bosonization of higher
dimensional Fermi surfaces introduced by Luther and advocated most recently by
Haldane. Bosonization incorporates the phase space and small-angle scattering
.... (7 figures, appended as a postscript file at the end of the TeX file).
