This paper focuses on a novel and challenging vision task, dense video captioning, which aims to automatically describe a video clip with multiple informative and diverse caption sentences. The proposed method is trained without explicit annotation of fine-grained sentence to video region-sequence correspondence, but is only based on weak video-level sentence annotations. It differs from existing video captioning systems in three technical aspects. First, we propose lexical fully convolutional neural networks (Lexical-FCN) with weakly supervised multi-instance multi-label learning to weakly link video regions with lexical labels. Second, we introduce a novel submodular maximization scheme to generate multiple informative and diverse region-sequences based on the Lexical-FCN outputs. A winner-takes-all scheme is adopted to weakly associate sentences to region-sequences in the training phase. Third, a sequence-to-sequence learning based language model is trained with the weakly supervised information obtained through the association process. We show that the proposed method can not only produce informative and diverse dense captions, but also outperform state-of-the-art single video captioning methods by a large margin.