The Chung–Lu network model is a multiplicative vertex weighted model, analogous to the Erdős–Rényi mixture model, in which the weights of the vertices may take arbitrary positive values. If the weights are restricted to being drawn from a fixed, finite set, the model is precisely equivalent to an Erdős–Rényi mixture model. Having arbitrary choices of weights allows rather general distributions for the number of neighbours of a vertex. In the Bernoulli random graph, this distribution is constrained to being close to Poisson, and, in particular, cannot have long tails, which makes unlikely that occasional vertices have very large numbers of neighbours, as is often observed in practice. The generality of the possible choices of weights makes the arguments more difficult than for Bernoulli models, since the approximation of the neighbourhoods is no longer as accurate. Despite this, it is possible to match the distribution of local neighbourhoods to those of a suitably chosen branching process, to prove a threshold theorem for the existence of a giant component and a subgraph threshold theorem for the numbers of small subgraphs, and to approximate the distribution of the shortest path length between two vertices having particular weights.