In many data-driven domains, a standard task is to understand how one distribution transforms into another on the basis of samples, with the goal of estimating this transformation for unseen out-of-sample points. The optimal transport map is one such canonical transformation between two distributions, which has been widely used in applied statistical problems, machine learning, and economics. Many estimators in the literature are prohibitively expensive and do not scale to settings where the number of samples is large or if the dimension is moderately large. To remedy both of these issues, we propose and analyze the entropic transport map, a computationally efficient estimator of the optimal transport map based on entropic optimal transport (Cuturi, 2013). Due to Sinkhorn's algorithm, we can take advantage of many samples for the purposes of estimation while maintaining a near-optimal convergence rate in the low-smoothness regime. Recently, we leveraged these results to provide statistical estimation rates for the Schrödinger bridge between two distributions. This is joint work with Jonathan Niles-Weed (NYU).