We present a framework for explaining variation in predator invasion success and predator impacts on native prey that integrates information about predatorprey naïveté, predator and prey behavioral responses to each other, consumptive and non‐consumptive effects of predators on prey, and interacting effects of multiple species interactions. We begin with the naïve prey hypothesis that posits that naïve, native prey that lack evolutionary history with non‐native predators suffer heavy predation because they exhibit ineffective antipredator responses to novel predators. Not all naïve prey, however, show ineffective antipredator responses to novel predators. To explain variation in prey response to novel predators, we focus on the interaction between prey use of general versus specific cues and responses, and the functional similarity of non‐native and native predators. Effective antipredator responses reduce predation rates (reduce consumptive effects of predators, CEs), but often also carry costs that result in non‐consumptive effects (NCEs) of predators. We contrast expected CEs versus NCEs for non‐native versus native predators, and discuss how differences in the relative magnitudes of CEs and NCEs might influence invasion dynamics. Going beyond the effects of naïve prey, we discuss how the naïve prey, enemy release and evolution of increased competitive ability (EICA) hypotheses are inter‐related, and how the importance of all three might be mediated by prey and predator naïveté. These ideas hinge on the notion that non‐native predators enjoy a novelty advantage associated with the naïveté of native prey and top predators. However, non‐native predators could instead suffer from a novelty disadvantage because they are also naïve to their new prey and potential predators. We hypothesize that patterns of community similarity and evolution might explain the variation in novelty advantage that can underlie variation in invasion outcomes. Finally, we discuss management implications of our framework, including suggestions for managing invasive predators, predator reintroductions and biological control.