Y chromosomes are genetically degenerate, having lost most of the active genes that were present in their ancestors. The causes of this degeneration have attracted much attention from evolutionary theorists. Four major theories are reviewed here: Muller's ratchet, background selection, the Hill Robertson effect with weak selection, and the 'hitchhiking' of deleterious alleles by favourable mutations. All of these involve a reduction in effective population size as a result of selective events occurring in a non-recombining genome, and the consequent weakening of the efficacy of selection. We review the consequences of these processes for patterns of molecular evolution and variation at loci on Y chromosomes, and discuss the results of empirical studies of these patterns for some evolving Y-chromosome and neo-Y-chromosome systems. These results suggest that the effective population sizes of evolving Y or neo-Y chromosomes are severely reduced, as expected if some or all of the hypothesized processes leading to degeneration are operative. It is, however, currently unclear which of the various processes is most important; some directions for future work to help to resolve this question are discussed.