Multi-Domain Neural Machine Translation (NMT) aims at building a single system that performs well on a range of target domains. However, along with the extreme diversity of cross-domain wording and phrasing style, the imperfections of training data distribution and the inherent defects of the current sequential learning process all contribute to making the task of multi-domain NMT very challenging. To mitigate these problems, we propose the Factorized Transformer, which consists of an in-depth factorization of the parameters of an NMT model, namely Transformer in this paper, into two categories: domain-shared ones that encode common cross-domain knowledge and domain-specific ones that are private for each constituent domain. We experiment with various designs of our model and conduct extensive validations on English to French open multi-domain dataset. Our approach achieves state-of-the-art performance and opens up new perspectives for multi-domain and open-domain applications.