A central and perhaps insurmountable challenge of invasion ecology is to predict which combinations of species and habitats most effectively promote and prevent biological invasions. Here, we integrate models of network structure and nonlinear population dynamics to search for potential generalities among trophic factors that may drive invasion success and failure. We simulate invasions where 100 different species attempt to invade 150 different food webs with 15-26 species and a wide range (0.06-0.32) of connectance. These simulations yield 11,438 invasion attempts by non-basal species, 47 per cent of which are successful. At the time of introduction, whether or not the invader is a generalist best predicts final invasion success; however, once the invader establishes itself, it is best distinguished from unsuccessful invaders by occupying a lower trophic position and being relatively invulnerable to predation. In general, variables that reflect the interaction between an invading species and its new community, such as generality and trophic position, best predict invasion success; however, for some trophic categories of invaders, fundamental species traits, such as having the centre of the feeding range low on the theoretical niche axis (for non-omnivorous and omnivorous herbivores), or the topology of the food web (for tertiary carnivores), best predict invasion success. Across all invasion scenarios, a discriminant analysis model predicted successful and failed invasions with 76.5 per cent accuracy for properties at the time of introduction or 100 per cent accuracy for properties at the time of establishment. More generally, our results suggest that tackling the challenge of predicting the properties of species and habitats that promote or inhibit invasions from food web perspective may aid ecologists in identifying rules that govern invasions in natural ecosystems.