The generalized homological mirror symmetry conjecture states that for mirror dual models $(X\_E, w)$ and $(X\_E', w')$ , if $L$, a lattice polytope which is a Newton polytope of a nonsingular projective toric variety, is in $M\_{R}$ , then isomorphisms exist between:
$D^b(X\_E,w)$ and $DFS(X\_E',w')$, $D^b(X\_E',w')$ and $DFS(X\_E,w)$, where $DFS$ is the derived Fukaya category and $D^b$ is derived category of coherent sheaves. 
What cases of this conjecture are open?
Any references are greatly appreciated.



Concerning the homological mirror symmetry conjecture
The conjecture has been solved for elliptic curves, abelian varieties, non-singular torus bundles over affine manifolds, and quartic surfaces. It remains to find a unification from algebraic geometry. A few references for this subject are the following:


Kontsevich, Maxim (1994), Homological algebra of mirror symmetry, arXiv:alg-geom/9411018.
Kontsevich, Maxim; Soibelman, Yan (2000), Homological Mirror Symmetry and torus fibrations, arXiv:math.SG/0011041.
Seidel, Paul (2003), Homological mirror symmetry for the quartic surface, arXiv:math.SG/0310414.
Hausel, Tamas; Thaddeus, Michael (2002), Mirror symmetry, Langlands duality, and the Hitchin system, arXiv:math.DG/0205236


