Identifying mix-and-match relationships between fashion items is an urgent task in a fashion e-commerce recommender system. It will significantly enhance user experience and satisfaction. However, due to the challenges of inferring the rich yet complicated set of compatibility patterns in a large e-commerce corpus of fashion items, this task is still underexplored. Inspired by the recent advances in multi-relational knowledge representation learning and deep neural networks, this paper proposes a novel Translation-based Neural Fashion Compatibility Modeling (TransNFCM) framework, which jointly optimizes fashion item embeddings and category-specific complementary relations in a unified space via an end-to-end learning manner. TransNFCM places items in a unified embedding space where a category-specific relation (category-comp-category) is modeled as a vector translation operating on the embeddings of compatible items from the corresponding categories. By this way, we not only capture the specific notion of compatibility conditioned on a specific pair of complementary categories, but also preserve the global notion of compatibility. We also design a deep fashion item encoder which exploits the complementary characteristic of visual and textual features to represent the fashion products. To the best of our knowledge, this is the first work that uses category-specific complementary relations to model the category-aware compatibility between items in a translation-based embedding space. Extensive experiments demonstrate the effectiveness of TransNFCM over the state-of-the-arts on two real-world datasets.