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 provide a limit theorem with rates of convergence for the distribution of a vertex, chosen uniformly at random, as the number of vertices tends to infinity. To do so, we develop Stein's method for a new class of limting distributions including power-laws. Similar, but slightly weaker results are shown to be deducible using coupling techniques. Concentrating on a specific preferential attachment model we also show that the outdegree distribution asymptotically follows a Poisson law. In addition, we deduce a central limit theorem for the number of isolated vertices. We thereto construct a size-bias coupling which in combination with Stein’s method also yields bounds on the distributional distance.