  I begin by explaining how Riemannian geometry can be understood in terms of
principal fibre bundles and connections thereon. I then introduce and motivate
the definition of a spinor structure in terms of familiar geometrical ideas.
  The central result of this thesis is a complete and constructive
classification of spinor structures, generalising some earlier results. I will
explain how principal fibre bundles and covering spaces provide the key
ingredients to the proof. A different type of classification can also be
attempted, in terms of the underlying principal fibre bundle, and this allows
us to compare `spinor connections'.
  The final part indicates how spinor structures for Lorentzian manifolds
provide the natural setting for the `spinor calculus', and so for the Dirac
equation for the electron. The effect of the choice of spinor structure on the
Dirac equation is investigated.
