Semantic specialization is a process of fine-tuning pre-trained distributional word vectors using external lexical knowledge (e.g., WordNet) to accentuate a particular semantic relation in the specialized vector space. While post-processing specialization methods are applicable to arbitrary distributional vectors, they are limited to updating only the vectors of words occurring in external lexicons (i.e., seen words), leaving the vectors of all other words unchanged. We propose a novel approach to specializing the full distributional vocabulary. Our adversarial post-specialization method propagates the external lexical knowledge to the full distributional space. We exploit words seen in the resources as training examples for learning a global specialization function. This function is learned by combining a standard L2-distance loss with a adversarial loss: the adversarial component produces more realistic output vectors. We show the effectiveness and robustness of the proposed method across three languages and on three tasks: word similarity, dialog state tracking, and lexical simplification. We report consistent improvements over distributional word vectors and vectors specialized by other state-of-the-art specialization frameworks. Finally, we also propose a cross-lingual transfer method for zero-shot specialization which successfully specializes a full target distributional space without any lexical knowledge in the target language and without any bilingual data.