Uphill diffusions in single and multi-species systems

Uphill diffusions constitute an intriguing phenomenon reported in a series of numerical simulations and experiments in which particles move from lower to higher density regions, at variance with the basic tenets of transport theory. In this paper we review several examples of uphill diffusions that appear in quite different frameworks. We highlight the role of the coupling with external reservoirs in the onset of particle currents with the 'wrong' sign, and also put forward a statistical mechanical explanation of the phenomenon for stochastic multi-species systems as well as for single-species models undergoing a phase transition.