Magnetic skyrmions are topologically protected spin configurations that have gained enormous interest as building blocks for next-generation spintronic devices. A distinctive feature of skyrmion systems is the topological Hall effect (THE): when electrons traverse a skyrmion, their trajectories bend in a spin-dependent fashion, generating a transverse Hall current. In a skyrmion crystal, this leads to electronic bands with non-trivial Chern numbers. In this talk, I will show how an externally applied electric field—via the Rashba spin–orbit interaction—provides a powerful tuning knob for the topological band properties. Specifically, we find that the Chern numbers of individual subbands can be modified by the field, driving a transition between an ordinary insulator and a Chern insulator. The Chern insulating state leads to an edge current in finite samples that can be manipulated by changing the topology of the bands.  Remarkably, for partially filled subbands, the Hall current can be reversed in sign at a critical electric-field strength, so that the current direction is flipped. These results suggest new strategies for manipulating charge transport in skyrmionic systems.