Electrical current as a means to control structural and related physical properties has been recognized only recently. The application of small electrical currents in sensitive detector and control applications, and in information technologies, is often preferable to other external stimuli. However, until recently it has not been widely accepted that electrical current can readily couple to the lattice, orbital, and spin degrees of freedom. Mounting experimental evidence has indicated that a combination of strong spin-orbit interactions and a distorted crystal structure in magnetic Mott insulators may be sufficient for electrical current to control structural and related properties. Current control of quantum states in 4d- and 5d-transition metal oxides has therefore rapidly expanded as a key research topic. This chapter presents two model systems, Ca₂RuO₄ and Sr₂IrO₄, in which applied current effectively controls the lattice, and thus the physical properties.