Ocean currents and atmospheric winds transport a variety of natural and man-made materials. Natural Lagrangian transports involve aerosols, pathogens, plankton, algae, sediments, and air and water masses themselves, while man-made quantities include pollutants, debris, floating objects, robots, or humans themselves in search and rescue operations. Due to advection and dynamical forces such as Coriolis, buoyancy, and atmospheric and tidal forcing, coherent structures such as fronts, jets, eddies, and gyres form. These coherent structures and their shear flows also lead to instabilities, turbulent stirring, and ultimately mixing and molecular diffusion. It is important for science and applications to predict, map, and characterize these transports and flow structures in space and time, and to differentiate advection, coherence, inertia, and reversible processes from instabilities, incoherence, mixing, and irreversible processes. In this work, we utilize our principled PDE-based methods for the probabilistic prediction, Bayesian estimation, optimal sampling, and machine learning of stochastic flow maps and Lagrangian transport in geophysical fluid flows.