Mesh-based simulations are central to modeling complex physical systems in many disciplines across science and engineering, as they support powerful numerical integration methods and their resolution can be adapted to strike favorable trade-offs between accuracy and efficiency. Here we introduce MeshGraphNets, a graph neural network-based method for learning simulations, which leverages mesh representations. Our model can be trained to pass messages on a mesh graph and to adapt the mesh discretization during forward simulation. We show that our method can accurately predict the dynamics of a wide range of physical systems, including aerodynamics, structural mechanics, and cloth-- and do so efficiently, running 1-2 orders of magnitude faster than the simulation on which it is trained. Our approach broadens the range of problems on which neural network simulators can operate and promises to improve the efficiency of complex, scientific modeling tasks.