Theory is equivocal about invasion success, implicating various combinations of r, a population's intrinsic rate of increase, K, its `carrying capacity', enemy-imposed death rates and the coefficient of variation in population numbers as determinants of establishment. Data for a wide variety of organisms, both vertebrates and invertebrates (including insects), accidentally or deliberately introduced by man into the British Isles, appear to show that the probability of establishment of an invader is positively correlated with body size. These data are consistent with the idea that the amplitude of population fluctuations is the main determinant of invasion success, but not with theoretically expected effects of r or K (because populations of small organisms generally fluctuate more than populations of large organisms, but both r and K decrease with increasing body size). However, data for various insect orders introduced into Britain show exactly the opposite trend, with probability of establishment decreasing with increasing body size, and hence possibly with decreasing r and/or K. Possible reasons for these contradictory results, including biases in the data, are discussed. Finally, data from a variety of sources, including insects released as biological control agents, show that enemies (parasitoids and predators) are often a cause of failure to establish particular species of introduced insects. These data add further to the uncertainty about the main determinants of invasion ability. The only clear conclusion to emerge from a combination of theoretical and empirical studies is that r alone is not obviously the main, or even an important determinant of invasion success.