Human conversations naturally evolve around related entities and connected concepts, while may also shift from topic to topic. This paper presents ConceptFlow, which leverages commonsense knowledge graphs to explicitly model such conversation flows for better conversation response generation. ConceptFlow grounds the conversation inputs to the latent concept space and represents the potential conversation flow as a concept flow along the commonsense relations. The concept is guided by a graph attention mechanism that models the possibility of the conversation evolving towards different concepts. The conversation response is then decoded using the encodings of both utterance texts and concept flows, integrating the learned conversation structure in the concept space. Our experiments on Reddit conversations demonstrate the advantage of ConceptFlow over previous commonsense aware dialog models and fine-tuned GPT -2 models, while using much fewer parameters but with explicit modeling of conversation structures. The rapid advancements of language modeling and natural language generation (NLG) techniques have enabled fully data-driven conversation models, which take user inputs (utterances) and directly generate natural language responses (Shang et al., 2015; Vinyals & Le, 2015; Li et al., 2016). On the other hand, the current generation models may still degenerate dull and repetitive contents (Holtz-man et al., 2019; Welleck et al., 2019), which, in conversation assistants, lead to irrelevant, off-topic, and non-useful responses that would damage user experiences (Tang et al., 2019; Zhang et al., 2018; Gao et al., 2019).