The volume entropy is a fundamental geometric invariant defined as the exponential growth rate of volumes of balls in the universal cover. It is a very subtle invariant which has attracted extensive study. The fundamental rigidity results here are the maximal volume entropy rigidity result of Ledrappier-Wang and the minimal volume rigidity theorem of Besson-Courtois-Gallot. The latter is a far reaching generalization of various famous rigidity results such as the Mostow rigidity for hyperbolic manifolds. We will report on joint work with Chris Connell, Xianzhe Dai, Jesus Nunez-Zimbron, Requel Perales, Pablo Suarez-Serrato concerning the generalizations to RCD spaces of these rigidity results.