Abstract: Motivated by matching and allocation problems we introduce the optimal transport problem between two invariant random measures. Since this is a transport problem between two infinite measures the total transport cost will always be infinite. It turns that the proper replacement is the transport cost per unit volume; assuming that the transport cost per unit volume is finite existence and uniqueness of optimal invariant couplings can be established. After reviewing the essential parts of this theory I will show that in dimension one there is a sharp threshold for the transport cost between the Lebesgue measure and an invariant random measure to be finite. More precisely, we show that the L^1 cost is always infinite (provided the random measure is sufficiently random) and we establish sharp and easily checkable conditions for the L^p cost to be finite for 0<p<1.

If time permits, we end with some challenging open problems.