Transport and mixing processes play an important role in many natural phenomena and their mathematical analysis has received considerable attention in the last two decades. The geometric approach to transport includes the study of invariant manifolds, which may act as barriers to particle transport and inhibit mixing. "Lagrangian coherent structures" have been introduced as finite-time proxies for invariant manifolds in non-autonomous settings. The ergodic-theoretic approach to transport includes the study of relaxation of initial ensemble densities to an invariant density, with a special focus on initial densities that relax more slowly than suggested by the rate of local trajectory separation. Such slowly decaying ensembles have been studied as "strange eigenmodes" in fluids and have been used to identify almost-invariant sets. I will describe numerical comparisons of the geometric and ergodic-theoretic approaches for a number of examples including fluid-like flow in two and three dimensions, dissipative flows in three dimensions, and a study of surface ocean flow in the Southern Ocean. I will also describe recent theoretical work on the development of ergodic-theoretic methods to handle non-autonomous systems.