There are several well-established books in the area of optimal transport (Villani, Optimal transport: old and new. Springer, 2009), gradient flows (Ambrosio et al., Gradient flows in metric spaces and in the space of probability measures. Birkhauser Verlag, 2008), and mean field control and games (Cardaliaguet et al., The master equation and the convergence problem in mean-field games, 2015) with computations and applications. Why do we need this lecture note for the Oberwolfach seminar 2023? We present a different angle or a shortcut from current books. The note follows a modern applied mathematics angle toward dynamical optimal transport and its generalizations. It presents some recent developments in related areas, written at an informal level. The note targets readers of high-year undergraduate students, graduate students, postdocs, and early career faculties. They may find some interesting problems and motivations to work in this area, where the technical part is left in the above-mentioned famous books. The note also presents some mathematics formulations and computational methods. This explains why optimal transport-related equations or algorithms are essential in mathematical data sciences. Towards this goal, we particularly prepare readers to digest some terminologies in modern data sciences, including tangent space, co-tangent space, Wasserstein metrics, Jordan–Kinderlehrer–Otto (JKO) schemes, generative models, neural ODEs, Deep JKO methods, and so on.