We will introduce some principal ideas behind Schubert calculus on the flag manifold, and specifically what they mean in the context of equivariant cohomology (which we will also introduce). We will show that Schubert calculus can be extended to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. We define a canonical class in the equivariant cohomology of the manifold M for each fixed point p in M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant Schubert classes. We show that the restriction of a canonical class associated to a fixed point p to fixed point q can be calculated by a rational function which depends only on the value of the moment map, and the restriction of other canonical classes to points of index exactly two higher. Therefore, the structure constants can be calculated by a similar rational function. Our restriction formula is *manifestly positive* in many cases, including when M is a flag manifold. Time permitting, we will show how we can strengthen these results when M is an *index-increasing* GKM manifold. This is joint work with Susan Tolman of University of IL, Champaign-Urbana.