  Let X be a T-variety, where T is an algebraic torus. We describe a fully
faithful functor from the category of T-equivariant vector bundles on X to a
certain category of filtered vector bundles on a suitable quotient of X by T.
We show that if X is factorial, this functor gives an equivalence of
categories. This generalizes Klyachko's description of equivariant vector
bundles on toric varieties. We apply our machinery to show that vector bundles
of low rank on projective space which are equivariant with respect to special
subtori of the maximal torus must split, generalizing a theorem of Kaneyama.
Further applications include descriptions of global vector fields on
T-varieties, and a study of equivariant deformations of equivariant vector
bundles.
