  The equivariant cohomology of a space with a group action is not only a ring
but also an algebra over the cohomology ring of the classifying space of the
acting group. We prove that toric manifolds (i.e. compact smooth toric
varieties) are isomorphic as varieties if and only if their equivariant
cohomology algebras are weakly isomorphic. We also prove that quasitoric
manifolds, which can be thought of as a topological counterpart to toric
manifolds, are equivariantly homeomorphic if and only if their equivariant
cohomology algebras are isomorphic.
