We study approximation algorithms and hardness of approximation for several versions of the problem of packing Steiner trees. For packing edge-disjoint Steiner trees of undirected graphs, we show APX-hardness for four terminals. For packing Steiner-node-disjoint Steiner trees of undirected graphs, we show a logarithmic hardness result, and give an approximation guarantee ofO (√n logn), wheren denotes the number of nodes. For the directed setting (packing edge-disjoint Steiner trees of directed graphs), we show a hardness result of Θ(m 1/3/−ɛ) and give an approximation guarantee ofO(m 1/2/+ɛ), wherem denotes the number of edges. We have similar results for packing Steiner-node-disjoint priority Steiner trees of undirected graphs.