The Schrödinger equation is usually thought of as governing the behaviour of matter on a small scale. By a small system may be meant anything from two particles up to a whole star. Here, I want to consider a slightly larger system, the Universe. As has been remarked elsewhere, Schrödinger's equation comes into its own when classical physics breaks down. An example of breakdown on a small scale was provided by the classical model of the atom. Classical physics predicted that the electron would spiral into the nucleus and matter would collapse. Indeed, quantum mechanics and Schrödinger's equation were invented precisely to overcome this problem. There is a similar problem with the Universe. Classical physics predicts that there was a time about ten billion years ago when the density of matter would have been infinite. This is called the Big Bang singularity, and most people take it to be the beginning of the Universe. However, here I want to report some recent work which shows that, if one applies the Schrödinger equation to the whole Universe, there is no singularity. Instead one gets a wave function which corresponds in a classical limit to a Universe which starts from a minimum radius, expands in an inflationary manner at first, goes over to a matter dominated expansion, reaches a maximum radius and collapses again.