In a dynamic world, mechanisms allowing prediction of future situations can provide a selective advantage. We suggest that memory systems differ in the degree of flexibility they offer for anticipatory behavior and put forward a corresponding taxonomy of prospection. The adaptive advantage of any memory system can only lie in what it contributes for future survival. The most flexible is episodic memory, which we suggest is part of a more general faculty of mental time travel that allows us not only (...) to go back in time, but also to foresee, plan, and shape virtually any specific future event. We review comparative studies and find that, in spite of increased research in the area, there is as yet no convincing evidence for mental time travel in nonhuman animals. We submit that mental time travel is not an encapsulated cognitive system, but instead comprises several subsidiary mechanisms. A theater metaphor serves as an analogy for the kind of mechanisms required for effective mental time travel. We propose that future research should consider these mechanisms in addition to direct evidence of future-directed action. We maintain that the emergence of mental time travel in evolution was a crucial step towards our current success. (shrink)