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We investigate the following question for machine translation (MT): can we develop a single universal MT model to serve as the common seed and obtain derivative and improved models on arbitrary language pairs? We propose mRASP, an approach to pre-train a universal multilingual neural machine translation model. Our key idea in mRASP is its novel technique of random aligned substitution, which brings words and phrases with similar meanings across multiple languages closer in the representation space. We pre-train a mRASP model on 32 language pairs jointly with only public datasets. The model is then fine-tuned on downstream language pairs to obtain specialized MT models. We carry out extensive experiments on 42 translation directions across a diverse settings, including low, medium, rich resource, and as well as transferring to exotic language pairs. Experimental results demonstrate that mRASP achieves significant performance improvement compared to directly training on those target pairs. It is the first time to verify that multiple low-resource language pairs can be utilized to improve rich resource MT. Surprisingly, mRASP is even able to improve the translation quality on exotic languages that never occur in the pre-training corpus. Code, data, and pre-trained models are available at https://github.com/linzehui/mRASP.