  In recent years, tremendous progress has been made on numerical algorithms
for solving partial differential equations (PDEs) in a very high dimension,
using ideas from either nonlinear (multilevel) Monte Carlo or deep learning.
They are potentially free of the curse of dimensionality for many different
applications and have been proven to be so in the case of some nonlinear Monte
Carlo methods for nonlinear parabolic PDEs.
  In this paper, we review these numerical and theoretical advances. In
addition to algorithms based on stochastic reformulations of the original
problem, such as the multilevel Picard iteration and the Deep BSDE method, we
also discuss algorithms based on the more traditional Ritz, Galerkin, and least
square formulations. We hope to demonstrate to the reader that studying PDEs as
well as control and variational problems in very high dimensions might very
well be among the most promising new directions in mathematics and scientific
computing in the near future.
