  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 we
review these progress and in particular that, 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, we introduce a new connection between our
topological picture and the well-known Marcus-Hush theory of electron transfer,
which we are able to connect with 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.
We finally report on a few recent applications to energy materials and
planetary sciences.
