Optimization and sampling are both of central importance in large-scale machine learning problems, but they are typically viewed as very different problems. This talk presents recent results that exploit the interplay between them. Viewing Markov chain Monte Carlo sampling algorithms as performing an optimization over the space of probability distributions, we demonstrate analogs of Nesterov’s acceleration approach in the sampling domain, in the form of a discretization of an underdamped Langevin diffusion. In the other direction, we view stochastic gradient optimization methods, such as those that are common in deep learning, as sampling algorithms, and study the finite-time convergence of their iterates to an invariant distribution.