Optimal transport provides a powerful variational framework for comparing probability distributions, but its most important objects are not only distances: they are also potentials and maps. This mini-course gives a self-contained introduction to optimal transport, with a focus on Kantorovich potentials, duality, and quantitative stability. After introducing the Monge and Kantorovich formulations, transport plans, duality, and optimal maps, the course will discuss recent quantitative stability results for potentials and maps. The emphasis will be on intuition, structural ideas, and applications to settings where transport maps are inferred from noisy, empirical, or discretized data. No prior knowledge of optimal transport will be assumed. The presentation will take a slightly nonstandard route. First, Kantorovich duality will be interpreted through the lens of tropical, or idempotent, probability: infima, suprema, and c-transforms appear as zero-temperature analogues of familiar probabilistic operations, while entropic regularization provides a soft version of the same structure. Second, optimal transport potentials will be compared with classical elliptic potentials, such as those associated with the Laplacian and the p-Laplacian. This analogy helps explain why stability of potentials and maps is a natural but delicate question.