High-speed locomotion is an essential survival strategy for animals, allowing populating harsh and unpredictable environments. Bio-inspired soft robots equally benefit from versatile and ultrafast motion but require appropriate driving mechanisms and device designs. Here, we present a class of small-scale soft electromagnetic robots made of curved elastomeric bilayers, driven by Lorentz forces acting on embedded printed liquid metal channels carrying alternating currents with driving voltages of several volts in a static magnetic field. Their dynamic resonant performance is investigated experimentally and theoretically. These robust and versatile robots can walk, run, swim, jump, steer and transport cargo. Their tethered versions reach ultra-high running speeds of 70 BL/s (body lengths per second) on 3D-corrugated substrates and 35 BL/s on arbitrary planar substrates while their maximum swimming speed is 4.8 BL/s in water. Moreover, prototype untethered versions run and swim at a maximum speed of 2.1 BL/s and 1.8 BL/s, respectively.