In the directed minimum spanning tree problem (DMST, also called minimum weight arborescence), the network is given a root node r, and needs to construct a minimum-weight directed spanning tree, rooted at r and oriented outwards. In this paper we present the first sub-quadratic DMST algorithms in the distributed CONGEST network model, where the messages exchanged between the network nodes are bounded in size. We consider three versions: a model where the communication links are bidirectional but can have different weights in the two directions; a model where communication is unidirectional; and the Congested Clique model, where all nodes can communicate directly with each other. Our algorithm is based on a variant of Lovász’ DMST algorithm for the PRAM model, and uses a distributed single-source shortest-path (SSSP) algorithm for directed graphs as a black box. In the bidirectional CONGEST model, our algorithm has roughly the same running time as the SSSP algorithm; using the state-of-the-art SSSP algorithm, we obtain a running time of Õ(Formula presented.) rounds for the bidirectional communication case. For the unidirectional communication model we give an Õ(n) algorithm, and show that it is nearly optimal. And finally, for the Congested Clique, our algorithm again matches the best known SSSP algorithm: it runs in Õ(n1/3) rounds. On the negative side, we adapt an observation of Chechik in the sequential setting to show that in all three models, the DMST problem is at least as hard as the (s,t)-shortest path problem. Thus, in terms of round complexity, distributed DMST lies between single-source shortest path and (s,t)-shortest path.