We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a (sub-)linear function of the indegree of the older vertex at that time. We develop Stein's method for the asymptotic power-law distribution of a vertex, chosen uniformly at random, and deduce rates of convergence as the number of vertices tends to \(\infty\). Using Stein's method for Poisson resp. Normal approximation we also show limit theorems for the outdegree distribution as well as for the number of isolated vertices. (Joint work with Hanna Döring and Marcel Ortgiese)