In recent years, Neural Machine Translation (NMT) has achieved notable results in various translation tasks. However, the word-by-word generation manner determined by the autoregressive mechanism leads to high translation latency of the NMT and restricts its low-latency applications. Non-Autoregressive Neural Machine Translation (NAT) removes the autoregressive mechanism and achieves significant decoding speedup by generating target words independently and simultaneously. Nevertheless, NAT still takes the word-level cross-entropy loss as the training objective, which is not optimal because the output of NAT cannot be properly evaluated due to the multimodality problem. In this article, we propose using sequence-level training objectives to train NAT models, which evaluate the NAT outputs as a whole and correlates well with the real translation quality. First, we propose training NAT models to optimize sequence-level evaluation metrics (e.g., BLEU) based on several novel reinforcement algorithms customized for NAT, which outperform the conventional method by reducing the variance of gradient estimation. Second, we introduce a novel training objective for NAT models, which aims to minimize the Bag-of-N-grams (BoN) difference between the model output and the reference sentence. The BoN training objective is differentiable and can be calculated efficiently without doing any approximations. Finally, we apply a three-stage training strategy to combine these two methods to train the NAT model. We validate our approach on four translation tasks (WMT14 En ↔ De, WMT16 En ↔ Ro), which shows that our approach largely outperforms NAT baselines and achieves remarkable performance on all translation tasks. The source code is available at https://github.com/ictnlp/Seq-NAT.