  It is well-known that perturbative calculations in field theory can lead to
far simpler answers than the Feynman diagram approach might suggest. In some
cases scattering amplitudes can be constructed for processes with any desired
number of external legs yielding compact expressions which are inaccessible
from the point of view of conventional perturbation theory. In this thesis we
discuss some attempts to address the nature of this underlying simplicity and
then use the results to calculate some previously unknown amplitudes of
interest. Witten's twistor string theory is introduced and the CSW rules at
tree-level and one-loop are described. We use these techniques to calculate the
one-loop gluonic MHV amplitudes in N=1 super-Yang-Mills as a verification of
their validity and then proceed to evaluate the general MHV amplitudes in pure
Yang-Mills with a scalar running in the loop. This latter amplitude is a new
result in QCD. In addition to this, we review some recent on-shell recursion
relations for tree-level amplitudes in gauge theory and apply them to gravity.
As a result we present a new compact form for the n-graviton MHV amplitudes in
general relativity. The techniques and results discussed are relevant to the
understanding of the structure of field theory and gravity and the
non-supersymmetric Yang-Mills amplitudes in-particular are pertinent to
background processes at the LHC. The gravitational recursion relations provide
new techniques for perturbative gravity and have some bearing on the
ultraviolet properties of Einstein gravity.
