A toric action on a symplectic manifold of dimension 2n is an
effective Hamiltonian action of T^n=(S^1)^n. Delzant's Theorem classifies
compact symplectic toric manifolds. It leaves open the following question:
Given a compact symplectic manifold, what toric actions does it admit?
How many, if any? We answer this question for manifolds of dimension four.
This is a joint work with Yael Karshon and Martin Pinsonnault.