We consider (probability-)measure valued processes, which we call MVMs, which have a natural martingale structure. Following previous work of Eldan and Cox-Kallblad, these pro- ¨ cesses are known to have a close connection to the solutions to the Skorokhod Embedding Problem. Here, we consider properties of these processes, and in particular, we are able to show that the MVMs connected to the Bass and Root embeddings have natural measure-valued analogues which also possess natural optimality properties. Based on joint work with M. Beiglbock, A.M.G. Cox and M. Huesmann.