  The appearance of a Hall conductance necessarily requires breaking of
time-reversal symmetry, either by an external magnetic field or by the internal
magnetization of a material. However, as a second response, Hall
dissipationless transverse currents can appear even in time-reversal symmetric
conditions in non-centrosymmetric materials. Moreover, this non-linear effect
has a quantum origin: it is related to the geometric properties of the
electronic wavefunctions and encoded in the dipole moment of the Berry
curvature. Here we review the general theory underpinning this effect and
discuss various material platforms where non-linear Hall transverse responses
have been theoretically proposed and experimentally verified. On the
theoretical front, the link between the non-linear Hall effect and the Berry
curvature dipole is discussed using Boltzmann transport theory. On the material
front, different platforms, including topological crystalline insulators,
transition metal dichalcogenides, graphene, and Weyl semimetals are reviewed.

Mesoscale and Nanoscale Physics