  This is a survey of the current state of the theory of $F$--(super)manifolds
$(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2],
[Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent
sheaf $\Cal{T}_M$, 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 quantum $K$--theory, theory
of extended moduli spaces, and unfolding spaces of singularities.
