For non-interacting fermions at zero temperature, it is well established that charge transport is quantized whenever the chemical potential lies in a gap of the single-body Hamiltonian. Proving the same result with interactions was an open problem for nearly 30 years until it was solved a few years ago by Hastings and Michalakis.  The solution uses new tools originally developed in the context of the classification of exotic phases of matter, and was used before in the proof of the many-dimensional Lieb-Schultz-Mattis theorem. I will explain these developments and show a theorem that unifies most of the known results on the subject. The same method applies also to FQHE and proves the existence of anyonic excitations in systems with fractional Hall conductance. The talk is based on a joint work with Alex Bols, Sven Bachmann and Wojciech De Roeck.

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