Simulation is important for countless applications in science and engineering, and there has been increasing interest in using machine learning for efficiency in prediction and optimization. In the first part of the talk, I will describe our work on training learned models for efficient turbulence simulation. Turbulent fluid dynamics are chaotic and therefore hard to predict, and classical simulators typically require expertise to produce and take a long time to run. We found that learned CNN-based simulators can learn to efficiently capture diverse types of turbulent dynamics at low resolutions, and that they capture the dynamics of a high-resolution classical solver more accurately than a classical solver run at the same low resolution. We also provide recommendations for producing stable rollouts in learned models, and improving generalization to out-of-distribution states. In the second part of the talk, I will discuss work using learned simulators for inverse design. In this work, we combine Graph Neural Network (GNN) learned simulators [Sanchez-Gonzalez et al 2020, Pfaff et al 2021] with gradient-based optimization in order to optimize designs in a variety of complex physics tasks. These include challenges designing objects in 2D and 3D to direct fluids in complex ways, as well as optimizing the shape of an airfoil. We find that the learned model can support design optimization across 100s of timesteps, and that the learned models can in some cases permit designs that lead to dynamics apparently quite different from the training data.