  In this review we show that a Clifford algebra possesses a unique irreducible
representation; the spinor representation. We discuss what types of spinors can
exist in Minkowski space-times and we explain how to construct all the
supersymmetry algebras that contain a given space-time Lie algebra. After
deriving the irreducible representations of the superymmetry algebras, we
explain how to use them to systematically construct supergravity theories. We
give the maximally supersymmetric supergravity theories in ten and eleven
dimensions and discuss their properties. We find which superbranes can exist
for a given supersymmetry algebra and we give the dynamics of the superbranes
that occur in M theory. Finally, we discuss how the properties of supergravity
theories and superbranes provide evidence for string duality.
  In effect, we present a continuous chain of argument that begins with
Clifford algebras and leads via supersymmetry algebras and their irreducible
representations to supergravity theories, string duality, brane dynamics and M
theory.
