Recent theoretical advances, based on a combination of concepts from Thouless' topological theory of adiabatic charge transport and a newly introduced gauge-invariance principle for transport coefficients, have permitted to connect (and reconcile) Faraday's picture of ionic transport-whereby each atom carries a well-defined integer charge-with a rigorous quantum description of the electronic charge-density distribution, which hardly suggests its partition into well defined atomic contributions. In this paper, these progresses are reviewed; in particular, it is shown how, by relaxing some general topological conditions, charge may be transported in ionic conductors without any net ionic displacements. After reporting numerical experiments which corroborate these findings, a new connection between the topological picture and the well-known Marcus-Hush theory of electron transfer is introduced in terms of the topology of adiabatic paths drawn by atomic trajectories. As a significant byproduct, the results reviewed here permit to classify different regimes of ionic transport according to the topological properties of the electronic structure of the conducting material. Finally, a few recent applications to energy materials and planetary sciences are reported.