In this paper, we propose Continuous Graph Flow, a generative continuous flow based method that aims to model complex distributions of graph-structured data. Our proposed model learns a joint probability density over a set of related random variables by formulating it as first order ordinary differential equation system with shared and reusable functions that operate over the graph structure. This leads to a reversible continuous message passing over time resulting in continuous transformations of probability distributions of the variables. We evaluate our model on a diverse set of generation tasks: graph generation, image puzzle generation, and layout generation from scene graphs. Experimental results show that CGF-based models outperform state-of-the-art graph generative models.