The invasion of ecosystems by nonnative species is widely considered the greatest threat to biodiversity after habitat loss. Given limited theoretical and empirical understanding of ecological robustness to such perturbations, we simulated invasions of complex ecological networks by integrating the ‘niche model’ of food web structure and a nonlinear bioenergetic model of population dynamics. Overall, 7958 successful invasions by 100 different invaders in 150 food webs with 15–29 original species (mean 20) and 5–38% connectance (mean 16%) showed that most (61%) communities were robust to invasion in that they experienced no species loss. The distribution of robustness in terms of the fraction of native species that persisted (mean 94%) was skewed with a long tail reaching to values as low as 20%. Loss of a single species occurred less frequently (14% of cases) than ‘extinction cascades’ involving the loss of two or more species (25% of cases). These cascades were often caused by invaders with many prey species and few predator species. While low-connectance webs and webs invaded by omnivores were most likely to lose at least one additional species, high-connectance webs experiencing extinction cascades lost the most species, especially when invaded by secondary consumers. These and earlier simulation results suggest how the structure of invaded communities and the properties of invaders involve trade-offs among robustness and resistance to invasion. For example, high-connectance communities are highly resistant and robust to invasion overall but lose the most species in the relatively few cases when extinctions occur. Low-connectance webs are the least resistant and more often lack robustness but lose the fewest species in the relatively many cases when extinctions occur. Broadly speaking, these findings suggest that high connectance makes food webs rigidly resistant to invasion but more brittle once such rigidity is breached. Low-connectance webs are less rigid while more flexibly suffering fewer extinctions when extinctions occur.