Over the past few decades, optimal transport theory has attracted growing interest across a range of fields, including partial differential equations, probability, and machine learning. Our discussion begins by introducing the basic theory of optimal transport. We then explore its diverse applications in various machine learning problems, with a particular focus on generative models. In addition, we examine the relationship between gradient flows and their discrete approximations, known as De Giorgi’s minimizing movements, in different spaces. Starting with the backward Euler method in Euclidean space, we will investigate gradient flows in the space of probability measures, equipped with the distance defined by the Monge-Kantorovich optimal transport problem. If time permits, we will also discuss recent advances in bilevel optimization and related open problems.