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The process of human adaptation to novel environments is a uniquely complex interplay between cultural and genetic changes. However, mechanistically, we understand little about these processes. To begin to untangle these threads of human adaptation we use mathematical models to describe and investigate cultural selective sweeps. We show that cultural sweeps differ in important ways from the genetic equivalents. The models show that the dynamics of cultural selective sweeps and, consequently, their differences from genetic sweeps depend critically on cultural transmission mechanisms. Further, we consider the effect of processes unique to culture such as foresight and innovations in response to an environmental change on adaptation. Finally we show that a ‘cultural evolutionary rescue’, or the survival of an endangered population by means of cultural adaptation, is possible. We suggest that culture …