  We review proofs of a theorem of Bloch on the absence of macroscopic
stationary currents in quantum systems. The standard proof shows that the
current in 1D vanishes in the large volume limit under rather general
conditions. In higher dimension, the total current across a cross-section does
not need to vanish in gapless systems but it does vanish in gapped systems. We
focus on the latter claim and give a self-contained proof motivated by a
recently introduced index for many-body charge transport in quantum lattice
systems having a conserved $U(1)$-charge.
