Weinan E - Princeton University, Mathematics In physics, economics, and control theory, we often encounter PDEs in very high dimensions. This has been a notoriously difficult problem due to the curse of dimensionality. In recent years, two classes of algorithms have emerged for solving nonlinear parabolic PDEs in high dimension, with a complexity that scales algebraically (linear or quadratic) in the dimension: the multi-level Picard method and the deep learning based methods. These algorithms have opened up new possibilities for attacking control and many other problems in hundreds and thousands of dimensions. They have also triggered questions about understanding PDEs in high dimensons. In this talk, I will discuss what we have achieved and understood so far about these problems.