Diamagnetic objects (polymer and metallic plates and spheres, ceramic beads, and liquid marbles), floating on water, and a variety of organic liquids may be driven by a steady magnetic field of 0.1 T, registered at the water-vapor surface. Diamagnetic bodies are attracted to the magnet, when the apparent contact angle at the solid/liquid interface is obtuse and repelled from the magnet, when the angle is acute. Cold plasma-treated polyolefin rafts and spheres, demonstrating underwater floating, are repelled by a permanent magnet. Addition of a surfactant to the water, as well as cold plasma treatment of the polyolefin bodies, can turn the attraction into the repulsion. We conjecture that the observed effects are caused by the interplay of two main phenomena. The first is the gravity, which induces sliding of the particle on the deformed liquid/vapor interface (the Moses effect). The second cause is the hysteresis of the contact angle at the bodies boundaries.