The problem of finding integer solutions to Diophantine equations is one that has fascinated mathematicians for thousands of years. Although we now know (thanks to the work of Davis, Matiyasevich, Putnam and Robinson resolving Hilbert's 10th problem in the negative) that it is impossible to do this in general, it ought to be possible to say a great deal in special cases. For example, when the equation in question defines an elliptic curve, a remarkable conjecture due to Birch and Swinnerton-Dyer implies that the behaviour of the solutions is governed by the properties of an analytic object (whose very existence is a deep problem in and of itself), namely the L-function attached to the elliptic curve. In this talk, I shall explain some of the ideas that go into the formulation of the Birch and Swinnerton-Dyer conjecture, and I shall discuss some aspects of what is currently known about the conjecture.