The co-evolution theory of relationship structure and interaction structure is an important issue in social networks. From this theory we can understand the change of relationship structure, evolution of interaction topology and the mutual influence between them, furthermore it can help us predict public sentiment and then control the network development. Because there are numerous interactions that could be observed and some real and latent relationships always cannot be gotten, inferring hidden relationship structure and predicting future relationships or actions become hot topics, which are also one way to uncover the principle of co-evolution. We summarize the research work of structure inference and prediction on social networks in this paper. We firstly give the definitions of co-evolution, structure inference and prediction, and analyze the relationship between them. Then we introduce and analyze the key technologies of structure inference and prediction. At last some challenges are given and future research topics are presented.