For next-generation technology, magnetic systems are of interest due to the natural ability to store information and, through spin transport, propagate this information for logic functions. Controlling the magnetization state through currents has proven energy inefficient. Multiferroic thin-film heterostructures, combining ferroelectric and ferromagnetic orders, hold promise for energy efficient electronics. The electric field control of magnetic order is expected to reduce energy dissipation by 2–3 orders of magnitude relative to the current state-of-the-art. The coupling between electrical and magnetic orders in multiferroic and magnetoelectric thin-film heterostructures relies on interfacial coupling though magnetic exchange or mechanical strain and the correlation between domains in adjacent functional ferroic layers. We review the recent developments in electrical control of magnetism through artificial magnetoelectric heterostructures, domain imprint, emergent physics and device paradigms for magnetoelectric logic, neuromorphic devices, and hybrid magnetoelectric/spin-current-based applications. Finally, we conclude with a discussion of experiments that probe the crucial dynamics of the magnetoelectric switching and optical tuning of ferroelectric states towards all-optical control of magnetoelectric switching events.