In this article, we aim at developing neighborhood-based neural models for link prediction. We design a novel multispace neighbor attention mechanism to extract universal neighborhood features by capturing latent importance of neighbors and selectively aggregate their features in multiple latent spaces. Grounded on this mechanism, we propose two link prediction models, i.e., self neighborhood attention network (SNAN), which predicts the link of two nodes by encoding and matching their respective neighborhood information, and its extension cross neighborhood attention network (CNAN), where we additionally design a cross neighborhood attention to directly capture structural interactions between two nodes. Another key novelty of this work is that we propose an adversarial learning framework, where a negative sample generator is devised to improve the optimization of the proposed link prediction models by continuously providing highly informative negative samples in the adversarial game. We evaluate our models with extensive experiments on 12 benchmark data sets against 14 popular and state-of-the-art link prediction approaches. The results strongly demonstrate the significant and universal superiority of our models on various types of networks. The effectiveness and robustness of the proposed attention mechanism and adversarial learning framework are also verified by detailed ablation studies.