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A common scenario of Multilingual Neural Machine Translation (MNMT) is that each translation task arrives in a sequential manner, and the training data of previous tasks is unavailable. In this scenario, the current methods suffer heavily from catastrophic forgetting (CF). To alleviate the CF, we investigate knowledge distillation based life-long learning methods. Specifically, in one-tomany scenario, we propose a multilingual distillation method to make the new model (student) jointly learn multilingual output from old model (teacher) and new task. In many-to one scenario, we find that direct distillation faces the extreme partial distillation problem, and we propose two different methods to address it: pseudo input distillation and reverse teacher distillation. The experimental results on twelve translation tasks show that the proposed methods can better consolidate the previous knowledge and sharply alleviate the CF.