Evolution in the merger rate as a function of redshift is in principle the key observable testing hierarchical models for the formation and evolution of galaxies. However, in practice, direct measurement of this quantity has proven difficult. In this opening review I outline the current best estimates for the merger rate as a function of cosmic epoch, focusing mostly upon recent advances made possible by deep ground-based redshift surveys and morphological studies undertaken with HST. I argue that a marriage of these techniques, in an attempt to determine the space density of mergers amongst the abundant morphologically peculiar population at high redshifts, is probably the most promising currently-available avenue for determining the prevalence of mergers at high redshifts. However, resolved kinematical studies, which seem set to become available in the next few years, are probably the best hope for a definitive determination of the space density of mergers at high redshifts.