In the mid 1970s a variation on Ornstein's d-bar distance yielded a variety of applications to source coding with a fidelity criterion, including a geometric description of the Shannon-distortion rate function, a measure of source coding mismatch, and an approach to   optimal rate-constrained simulation of stationary and ergodic sources using stationary codings of iid processes.  Finite dimensional versions of the distance date back to the late eighteenth century introduction of optimal transport theory by Monge and its mid-twentieth century development in an economics context by Kantorovich. The field of optimal transport is currently enjoying something of a renaissance in mathematics.  The goal of this talk is to highlight a few connections among optimal transport and   source coding, including some extensions of old and recent results along with a few conjectures.