I will discuss adapted transport theory and the adapted Wasserstein distance, which extend classical transport theory from probability measures to stochastic processes by incorporating the temporal flow of information. This adaptation addresses key limitations of classical transport when dealing with time-dependent data. 

I will highlight how, unlike other topologies for stochastic processes, the adapted Wasserstein distance ensures continuity for fundamental probabilistic operations, including the Doob decomposition, optimal stopping, and stochastic control. Additionally, I will explore how adapted transport preserves many desirable properties of classical transport theory, making it a powerful tool for analyzing stochastic systems.