Modern data science problems related to detection, estimation, clustering, and classification using data emanating from physical sensors (e.g. signals and images ) often pose difficult challenges due to nonlinearities present in complex phenomena. When data is generated from processes related to transport phenomena, solutions based on optimal transport and other Lagrangian embeddings capable of yielding high accuracy solutions for low computational cost have emerged. In this mini-course we will 1) present an overview of mathematical principles related to optimal transport, 2) define mathematical signal/image transforms based on transport and optimal transport, and 3) describe the application of these techniques towards modern problems in data science.