Motivated by the question of how macromolecules assemble, the notion of an assembly tree of a graph is introduced. Given a graph G, the paper is concerned with enumerating the number of assembly trees of G, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call (H, φ)-graphs. In some natural special cases, we apply powerful recent results of Zeilberger and Apagodu on multivariate generating functions, and results of Wimp and Zeilberger, to deduce recurrence relations and very precise asymptotic formulas for the number of assembly trees of the complete bipartite graphs K_n,n and the complete tripartite graphs K_n,n,n. Future directions for reseach, as well as open questions, are suggested