The correlations between the degrees on both ends of a randomly selected edge, called degree-degree correlations or network assortativity, are an important second order characterization of the structure of networks. In this talk I will discuss the behavior of these correlations in two different network models, one with neutral mixing and one which is maximally disassortative. The first is the directed configuration model, where the degree-degree correlations converge to zero. Here we will see that when the degree distribution has infinite variance we encounter negative correlations, generated by finite size effects. I will explain this phenomenon and discuss some scaling results for these negative correlations, in terms of the size of the network. The second model is the Dissassortative Graph Model, which was recently developed by myself and co-authors and designed to minimize degree-degree correlations. I will explain the joint degree structure of these maximally disassortative graphs which in turn determine the value of the degree-degree correlations. In addition, I will use this model to show that, although the absolute minimal correlation value is -1, the minimal achievable value on any network depends strongly on the degree distribution and converges to zero as the tail of the distribution decreases.