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Link prediction aims to infer the link existence between pairs of nodes in networks/graphs. Despite their wide application, the success of traditional link prediction algorithms is hindered by three major challenges -- link sparsity, node attribute noise and dynamic changes -- that are faced by many real-world networks. To address these challenges, we propose a Contextualized Self-Supervised Learning (CSSL) framework that fully exploits structural context prediction for link prediction. The proposed CSSL framework learns a link encoder to infer the link existence probability from paired node embeddings, which are constructed via a transformation on node attributes. To generate informative node embeddings for link prediction, structural context prediction is leveraged as a self-supervised learning task to boost the link prediction performance. Two types of structural context are investigated, i.e., context nodes collected from random walks vs. context subgraphs. The CSSL framework can be trained in an end-to-end manner, with the learning of model parameters supervised by both the link prediction and self-supervised learning tasks. The proposed CSSL is a generic and flexible framework in the sense that it can handle both attributed and non-attributed networks, and operate under both transductive and inductive link prediction settings. Extensive experiments and ablation studies on seven real-world benchmark networks demonstrate the superior performance of the proposed self-supervision based link prediction algorithm over state-of-the-art baselines, on different types of networks under both transductive and inductive settings. The proposed CSSL also yields competitive performance in terms of its robustness to node attribute noise and scalability over large-scale networks.