  We develop and study a theory of optimal transport for vector measures. We
resolve in the negative a conjecture of Klartag, that given a vector measure on
Euclidean space with total mass zero, the mass of any transport set is again
zero. We provide a counterexample to the conjecture. We generalise the
Kantorovich--Rubinstein duality to the vector measures setting. Employing the
generalisation, we answer the conjecture in the affirmative provided there
exists an optimal transport with absolutely continuous marginals of its total
variation.
