  In the main part of this thesis, we present the foundations and initial
results of the Spinorial Geometry formalism for solving Killing spinor
equations. This method can be used for any supergravity theory, although we
largely focus on D=11 supergravity. The D=5 case is investigated in an
appendix. The exposition provides a comprehensive introduction to the
formalism, and contains background material on the complex spin representations
which, it is hoped, will provide a useful bridge between the mathematical
literature and our methods. Many solutions to the D=11 Killing spinor equations
are presented, and the consequences for the spacetime geometry are explored in
each case. Also in this thesis, we consider another class of supergravity
solutions, namely heterotic string backgrounds with (2,0) world-sheet
supersymmetry. We investigate the consequences of taking alpha-prime
corrections into account in the field equations, in order to remain consistent
with anomaly cancellation, while requiring that spacetime supersymmetry is
preserved.
