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Current recommender systems consider the various aspects of items for making accurate recommendations. Different users place different importance to these aspects which can be thought of as a preference/attention weight vector. Most existing recommender systems assume that for an individual, this vector is the same for all items. However, this assumption is often invalid, especially when considering a user's interactions with items of diverse characteristics. To tackle this problem, in this paper, we develop a novel aspect-aware recommender model named A3NCF, which can capture the varying aspect attentions that a user pays to different items. Specifically, we design a new topic model to extract user preferences and item characteristics from review texts. They are then used to 1) guide the representation learning of users and items, and 2) capture a user's special attention on each aspect of the targeted item with an attention network. Through extensive experiments on several large-scale datasets, we demonstrate that our model outperforms the state-of-the-art review-aware recommender systems in the rating prediction task.