Unified string theories have led to new partial differential equations, many requiring a novel and unfamiliar geometry. Calabi-Yau manifolds were the earliest example, but many other equations have emerged since, each of which can be interpreted as defining a new notion of canonical metric in a non-Kaehler setting. Because of cohomological constraints, geometric flows seem particularly appropriate for the study of these equations. We provide a survey of them, with emphasis on the many open questions.
