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 study the geometry of the Grassmannians of symplectic subspaces in a symplectic
vector space. We construct symplectic twistor spaces by the symplectic quotient
construction and use them to describe the symplectic geometry of the symplectic Grassmannians.