This thesis studies new frameworks and methods for infinite-dimensional statistical inference and learning problems through the lens of optimal transport. Optimal transport is a branch of mathematics concerning how to transport one probability distribution to another while minimizing certain costs, which, as this thesis shows, can provide principled guidance for sophisticated modern data science problems. There are two overarching themes in this thesis: (1) the dynamic viewpoint of optimal transport that concerns the optimal path of transporting one probability distribution to another, and (2) quadratic extensions of the classical linear optimal transport problem. This thesis effectively utilizes these themes to delve into central problems in modern data science, including hypothesis testing, online learning, simulation-based inference, and missing value imputation. The results of this thesis contribute to the growing body of literature on optimal transport and its applications in statistics and machine learning, providing new insights and tools for researchers and practitioners in these fields.