  We investigate the convergence of the derivative expansion of the exact
renormalisation group, by using it to compute the beta function of scalar
theory. We demonstrate that the derivative expansion of the Polchinski flow
equation converges at one loop for certain fast falling smooth cutoffs. The
derivative expansion of the Legendre flow equation trivially converges at one
loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale
expansion), and rapidly in the case of a smooth exponential cutoff. We also
show that the two loop contributions to certain higher derivative operators
(not involved in beta) have divergent momentum-scale expansions for sharp
cutoff, but the smooth exponential cutoff gives convergent derivative
expansions for all such operators with any number of derivatives. In the latter
part of the thesis, we address the problems of applying the exact
renormalisation group to gauge theories. A regularisation scheme utilising
higher covariant derivatives and the spontaneous symmetry breaking of the gauge
supergroup SU(N|N) is introduced and it is demonstrated to be finite to all
orders of perturbation theory.
