In this talk I will present some recent work on the online version of the problem. Namely, requests appear in a sequence, and we want to maintain a Steiner arborescence of small cost. The analysis appeals to the concept of the edge-asymmetry of a graph, defined as the maximum ratio of the weights of antiparallel edges in the graph.
I will also outline some related variations of the Steiner tree problem which, likewise, apply in the on-line setting.