  We use Stein's method to obtain distributional approximations of subgraph
counts in the uniform attachment model or random directed acyclic graph; we
provide also estimates of rates of convergence. In particular, we give uni- and
multi-variate Poisson approximations to the counts of cycles, and normal
approximations to the counts of unicyclic subgraphs; we also give a partial
result for the counts of trees. We further find a class of multicyclic graphs
whose subgraph counts are a.s. bounded as $n\to\infty$.
