Almost toric manifolds form a class of singular Lagrangian fibered
symplectic manifolds that include both toric manifolds and the K3
surface. We classify closed almost toric four-manifolds up to diffeomorphism
and indicate precisely the structure of all almost toric fibrations
of closed symplectic four-manifolds. A key step in the proof is a geometric
classification of the singular integral affine structures that can
occur on the base of an almost toric fibration of a closed four-manifold.
As a byproduct we provide a geometric explanation for why a generic
Lagrangian fibration over the two-sphere must have 24 singular fibers.
We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.