Several geometric and probabilistic methods for studying chaotic phase space transport have been developed and fruitfully applied to diverse areas from orbital mechanics to fluid mechanics and beyond. Increasingly, systems of interest are determined not by analytically defined model systems, but by data from experiments or large-scale simulations. This emphasis on real-world systems sharpens our focus on those features of phase space transport in finite-time systems which seem robust, leading to the consideration of not only invariant manifolds (separatrices) and invariant manifold-like objects, but also their connection with concepts such as symbolic dynamics, coherent sets, and optimal control. We will highlight some recent applications to areas such as spacecraft trajectories, microfluidic mixing, ship capsize prediction, biological invasions of airborne pathogens, and the gliding behavior of flying snakes.