Applied optimal transport is flourishing after computational advances have enabled its use in real-world problems with large data sets. Entropic regularization is a key method to approximate optimal transport in high dimensions while retaining feasible computational complexity. In this talk we introduce the basics of entropic optimal transport and touch on several recent developments: convergence rates for Sinkhorn's algorithm, connections with the rate–distortion tradeoff in compression theory, and selection of optimal transports for vanishing regularization.