We will discuss certain new directions in the nexus of ideas that originate in Optimal Mass Transport (OMT) and the Schroedinger Bridge Problem (SBP). We will begin with a brief historic overview, explain the relation between OMT and SBP, discuss applications in control, physics, and networks,  and we will conclude with generalizations to the setting of matrix-valued and vector-valued distributions. This final chapter is pertinent to quantum mechanics as it explains the Lindblad equation of open quantum systems as gradient flow of the von Neumann entropy, and it is pertinent to multivariable signal and image processing (DTI, color, etc.). 

The talk is based on joint work with Yongxin Chen (ISU), Michele Pavon (University of Padova), and Allen Tannenbaum (Stony Brook).