Transport barriers organize material distribution, pattern formation, and transport pathways in moving continua, ranging from planar steady flows to three-dimensional turbulence. There is a pressing need to identify such barriers objectively in geophysical flow data. Applications include tracking environmental contamination, forecasting clear-air turbulence for airports, and quantifying long-range eddy transport for climate-change predictions. Here we describe an emerging mathematical theory of transport barriers that provides an extension of well-known invariant sets in simple dynamical systems to complex, finite-time continuum dynamics. We also show how this approach accurately uncovers barriers in time-resolved oceanic and atmospheric data sets.