  It is well-known that a non-vanishing Hall conductivity requires
time-reversal symmetry breaking. However, in this work, we demonstrate that a
Hall-like transverse current can occur in second-order response to an external
electric field in a wide class of time-reversal invariant and inversion
breaking materials, at both zero and twice the optical frequency. This
nonlinear Hall effect has a quantum origin arising from the dipole moment of
the Berry curvature in momentum space, which generates a net anomalous velocity
when the system is in a current-carrying state. We show that the nonlinear Hall
coefficient is a rank-two pseudo-tensor, whose form is determined by point
group symmetry. We discus optimal conditions to observe this effect and propose
candidate two- and three-dimensional materials, including topological
crystalline insulators, transition metal dichalcogenides and Weyl semimetals.
