Recently, neural sequence-to-sequence models have made impressive progress in abstractive document summarization. Unfortunately, as neural abstractive summarization research is in a primitive stage, the performance of these models is still far from ideal. In this paper, we propose a novel method called Neural Abstractive Summarization with Diverse Decoding (NASDD). This method augments the standard attentional sequence-to-sequence model in two aspects. First, we introduce a diversity-promoting beam search approach in the decoding process, which alleviates the serious diversity issue caused by standard beam search and hence increases the possibility of generating summary sequences that are more informative. Second, we creatively utilize the attention mechanism combined with the key information of the input document as an estimation of the salient information coverage, which aids in finding the optimal summary sequence. We carry out the experimental evaluation with state-of-the-art methods on the CNN /Daily Mail summarization dataset, and the results demonstrate the superiority of our proposed method.