Flow-based models have spurred a revolution in generative modeling, driving astounding advancements across diverse domains including high-resolution text to image synthesis and de-novo drug design. Yet despite their remarkable performance, inference in these models requires the solution of a differential equation, which is extremely costly for the large-scale neural network-based models used in practice. In this talk, we introduce a mathematical theory of flow maps, a new class of generative models that directly learn the solution operator for a flow-based model. By learning this operator, flow maps can generate data in 1-4 network evaluations, leading to orders of magnitude faster inference compared to standard flow-based models. We discuss several algorithms for efficiently learning flow maps in practice that emerge from our theory, and we show how many popular recent methods for accelerated inference -- including consistency models, shortcut models, and mean flow -- can be viewed as particular cases of our formalism. We demonstrate the practical effectiveness of flow maps across several tasks including image synthesis, geometric data generation, and inference-time guidance of pre-trained text-to-image models.