John D Barrow gave the Gerald Whitrow Lecture for 2002. He reviews modern ideas about the Big Bang and the constants of Nature. Abstract The modern picture of the expanding Big Bang universe is described. Implications of the expansion for the evolution of life are highlighted, together with the new features contributed by the inflationary universe theory. Observational tests of inflation are described along with some of the possibilities introduced by new theories of strings and quantum gravity. These theories allow the numbers of dimensions of space and of time to be larger than the three and one we experience and permit the observed “constants” of Nature to vary slowly in time. We describe recent astronomical evidence that is consistent with small variations of the fine-structure constant and discuss some of its far-reaching implications.