Several geometric and probabilistic methods for studying chaotic phase space transport have been developed and fruitfully applied to diverse areas from orbital mechanics to biomechanics 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. We will discuss those features of phase space transport in finite-time systems which seem to be robust, considering invariant manifolds and invariant manifold-like objects, and their connection with concepts such as symbolic dynamics, chaos, almost-invariant sets, and coherent sets. We end with an application to efficient movement of weakly propelled vehicles in fluid flows.