  This work continues the study of $F$--manifolds $(M,\circ)$, first defined by
Hertling and Manin and investigated in [He]. The notion of a compatible flat
structure $\nabla$ is introduced, and it is shown that many constructions known
for Frobenius manifolds do not in fact require invariant metrics and can be
developed for all such triples $(M,\circ ,\nabla).$ In particular, we extend
and generalize recent Dubrovin's duality.
