Random dynamical systems are flexible mathematical models for the study of complicated systems whose evolution is affected by external factors, such as seasonal influences and random effects. Multiplicative ergodic theory provides fundamental information for the study of transport phenomena in such systems, including long-term behaviour, mixing rates and coherent structures. In this talk, we will present recent developments on random dynamical systems and multiplicative ergodic theory, guided in part by questions arising from the investigation of geophysical flows.