  Information geometry and optimal transport are two distinct geometric
frameworks for modeling families of probability measures. During the recent
years, there has been a surge of research endeavors that cut across these two
areas and explore their links and interactions. This paper is intended to
provide an (incomplete) survey of these works, including entropy-regularized
transport, divergence functions arising from $c$-duality, density manifolds and
transport information geometry, the para-K\"ahler and K\"ahler geometries
underlying optimal transport and the regularity theory for its solutions. Some
outstanding questions that would be of interest to audience of both these two
disciplines are posed. Our piece also serves as an introduction to the Special
Issue on Optimal Transport of the journal Information Geometry.
