Correspondence problems involving matching of two or more geometric domains find application across disciplines, from machine learning to computer vision.  A theoretical framework involving correspondence along geometric domains is optimal transport (OT). Dating back to early economic applications, the OT problem has received renewed interest thanks to its applicability to correspondence problems machine learning, computer graphics, geometry, and other disciplines.  The main barrier to wide adoption of OT as a modeling tool is the expense of optimization in OT problems.  In this talk, I will summarize efforts in my group to make large-scale transport tractable over a variety of domains and in a variety of application scenarios, helping transition OT from theory to practice.  In addition, I will show how OT can be used as a unit in systems for correspondence problems involving the processing of geometrically-structured data.