It is widely believed that Laurent Schwartz showed that it was impossible to multiply distributions. However in the 1980's J F Colombeau constructed a commutative and associative differential algebra for which there was a canonical embedding of the space of distributions as a linear subspace and a canonical embedding of the space of smooth functions as a subalgebra. I will start by outlining this construction and explain how it gets round the Schwartz "impossibility" result. Unfortunately the Colombeau construction relies on the linear structure of R^n. I will show how (in joint work with a group of mathematicians in Vienna) it was possible to reformulate the theory to allow the multiplication of distributions on manifolds. I will also explain how the theory has been extended to provide a theory of distributional differential geometry. I will end by giving some applications of these ideas to general relativity, firstly to give a description of weak singularities and secondly to obtain solutions of the wave equation on singular spacetimes.