The process by which a species becomes a biological invader, at a location where it does not naturally occur, can be divided into a series of sequential stages (transport, introduction, establishment and spread). A species' success at passing through each of these stages depends, in a large part, on the number of individuals available to assist making each transition. Here, we review the evidence that numbers determine success at each stage of the invasion process and then discuss the likely mechanisms by which numbers affect success. We conclude that numbers of individuals affect transport and introduction by moderating the likelihood that abundant (and widespread) species are deliberately or accidentally translocated; affect establishment success by moderating the stochastic processes (demographic, environmental, genetic or Allee) to which small, introduced populations will be vulnerable; and affect invasive spread most likely because of persistent genetic effects determined by the numbers of individuals involved in the establishment phase. We finish by suggesting some further steps to advance our understanding of the influence of numbers on invasion success, particularly as they relate to the genetics of the process.