Informally, a preferential attachment scheme is a dynamic (reinforcement) process where a network is grown by attaching new vertices to old ones selected with probability proportional to their weight, a function of their degrees. In these two talks, we give an introduction to these schemes and discuss a couple of approaches to the study of the large scale degree structure of these graphs. One fruitful approach is to view the scheme in terms of branching processes. Another is to understand it in terms of Markov decompositions and fluid limits.