We present Attract-Repel, an algorithm for improving the semantic quality of word vectors by injecting constraints extracted from lexical resources. Attract-Repel facilitates the use of constraints from mono- and cross-lingual resources, yielding semantically specialized cross-lingual vector spaces. Our evaluation shows that the method can make use of existing cross-lingual lexicons to construct high-quality vector spaces for a plethora of different languages, facilitating semantic transfer from high- to lower-resource ones. The effectiveness of our approach is demonstrated with state-of-the-art results on semantic similarity datasets in six languages. We next show that Attract-Repel-specialized vectors boost performance in the downstream task of dialogue state tracking (DST) across multiple languages. Finally, we show that cross-lingual vector spaces produced by our algorithm facilitate the training of multilingual DST models, which brings further performance improvements.