This talk reviews the current state of the theory of F–(super)manifolds (M, ◦), first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here ◦ is an OM–bilinear multiplication on the tangent sheaf TM , satisfying an integrability condition. F–manifolds and compatible flat structures on them furnish a useful weakening of Dubrovin’s Frobenius structure which naturally arises in the q...