  We introduce Wilson's, or Polchinski's, exact renormalization group, and
review the Local Potential Approximation as applied to scalar field theory.
Focusing on the Polchinski flow equation, standard methods are investigated,
and by choosing restrictions to some sub-manifold of coupling constant space we
arrive at a very promising variational approximation method. Within the Local
Potential Approximation, we construct a function, C, of the coupling constants;
it has the property that (for unitary theories) it decreases monotonically
along flows and is stationary only at fixed points - where it `counts degrees
of freedom', i.e. is extensive, counting one for each Gaussian scalar. In the
latter part of the thesis, the Local Potential Approximation is used to derive
a non-trivial Polchinski flow equation to include Fermi fields. Our flow
equation does not support chirally invariant solutions and does not reproduce
the features associated with the corresponding invariant theories. We solve
both for a finite number of components, N, and within the large N limit. The
Legendre flow equation provides a comparison with exact results in the large N
limit. In this limit, it is solved to yield both chirally invariant and
non-invariant solutions.
