Explore a comprehensive mini-tutorial on computational optimal transport, blending recent techniques to solve complex problems. Begin with the fundamentals of Monge and Kantorovich problems before delving into advanced methods such as convex optimization, automatic differentiation, and neural networks. Learn how these approaches can be applied to approximate optimal transport at scale, handle various cost structures, and operate in high-dimensional regimes. Gain insights from Marco Cuturi of Apple, Inc., who presented this talk at the 2022 SIAM Conference on Mathematics of Data Science in San Diego, California.