The notion of a virtual Woodin cardinal is obtained, roughly speaking, by weakening the definition of Woodin cardinal to allow generic elementary embeddings in place of elementary embeddings. As usual, the virtualization is much weaker than the original: if zero sharp exists then every Silver indiscernible is virtually Woodin in L. The least virtual Woodin cardinal is notable as the least cardinal such that every tree of that cardinality without infinite branches is isomorphic to a proper subtree of itself. We characterize virtual Woodin cardinals by a partition relation that is obtained by weakening the definition of Erdos cardinal to require, roughly speaking, many finite homogeneous sets instead of an infinite homogeneous set. We also define n-fold variants of virtual Woodin cardinals that are interleaved in consistency strength with the n-iterable cardinals and can also be characterized by partition relations.