As the fraction of electricity generated from intermittent renewable sources (such as solar and wind) grows, developing reliable energy storage technologies to store electrical energy in large scale is of increasing importance. Redox flow batteries are now enjoying a renaissance and regarded as a leading technology in providing a well-balanced solution for current daunting challenges. In this article, state-of-the-art studies of the complex multicomponent transport phenomena in aqueous redox flow batteries, with a special emphasis on all-vanadium redox flow batteries, are reviewed and summarized. Rather than elaborating on the details of previous experimental and numerical investigations, this article highlights: i) the key transport issues in each battery's component that need to be tackled so that the rate capability and cycling stability of flow batteries can be significantly improved, ii) the basic mechanisms that control the active species/ion/electron transport behaviors in each battery's component, and iii) the key experimental and numerical findings regarding the correlations between the multicomponent transport processes and battery performance.