Magnetic devices are a leading contender for the implementation of memory and logic technologies that are non-volatile, that can scale to high density and high speed, and that do not wear out. However, widespread application of magnetic memory and logic devices will require the development of efficient mechanisms for reorienting their magnetization using the least possible current and power(1). There has been considerable recent progress in this effort; in particular, it has been discovered that spin-orbit interactions in heavy-metal/ferromagnet bilayers can produce strong current-driven torques on the magnetic layer(2-11), via the spin Hall effect(12,13) in the heavy metal or the Rashba-Edelstein effect(14,15) in the ferromagnet. In the search for materials to provide even more efficient spin-orbit-induced torques, some proposals(16-19) have suggested topological insulators(20,21), which possess a surface state in which the effects of spin-orbit coupling are maximal in the sense that an electron's spin orientation is fixed relative to its propagation direction. Here we report experiments showing that charge current flowing in-plane in a thin film of the topological insulator bismuth selenide (Bi2Se3) at room temperature can indeed exert a strong spin-transfer torque on an adjacent ferromagnetic permalloy (Ni81Fe19) thin film, with a direction consistent with that expected from the topological surface state. We find that the strength of the torque per unit charge current density in Bi2Se3 is greater than for any source of spin-transfer torque measured so far, even for non-ideal topological insulator films in which the surface states coexist with bulk conduction. Our data suggest that topological insulators could enable very efficient electrical manipulation of magnetic materials at room temperature, for memory and logic applications.