  Excitons are neutral objects, that, naively, should have no response to a
uniform, electric field. Could the Berry curvature of the underlying electronic
bands alter this conclusion? In this work, we show that Berry curvature can
indeed lead to anomalous transport for excitons in 2D materials subject to a
uniform, in-plane electric field. By considering the constituent electron and
hole dynamics, we demonstrate that there exists a regime for which the
corresponding anomalous velocities are in the same direction. We establish the
resulting center of mass motion of the exciton through both a semiclassical and
fully quantum mechanical analysis, and elucidate the critical role of Bloch
oscillations in achieving this effect. We identify transition metal
dichalcogenide heterobilayers as candidate materials to observe the effect.
