The spin Hall effect (SHE) generates spin currents within nonmagnetic materials. Previously, studies of the SHE have been motivated primarily to understand its fundamental origin and magnitude. Here we demonstrate, using measurement and modeling, that in a Pt/Co bilayer with perpendicular magnetic anisotropy the SHE can produce a spin transfer torque that is strong enough to efficiently rotate and reversibly switch the Co magnetization, thereby providing a new strategy both to understand the SHE and to manipulate magnets. We suggest that the SHE torque can have a similarly strong influence on current-driven magnetic domain wall motion in Pt/ferromagnet multilayers. We estimate that in optimized devices the SHE torque can switch magnetic moments using currents comparable to those in magnetic tunnel junctions operated by conventional spin-torque switching, meaning that the SHE can enable magnetic memory and logic devices with similar performance but simpler architecture than the current state of the art.