We survey the recent work on phase transition and distances in various random graph models with general degree sequences. We focus on inhomogeneous random graphs, the configuration model, and affine preferential attachment models, and pay special attention to the setting where these random graphs have a power-law degree sequence. This means that the proportion of vertices with degree k in large graphs is approximately proportional to k- for some >1. Since many real networks have been empirically shown to have power-law degree sequences, these random graphs can be seen as more realistic models for real complex networks than classical random graphs such as the Erdos–Rényi random graph. It is often suggested that the behavior of random graphs should have a large amount of universality, meaning, in this case, that random graphs with similar degree sequences share similar behavior. We survey the available results on graph distances in power-law random graphs that are consistent with this prediction.