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Spin-orbit torques (SOT) provide a versatile tool to manipulate the magnetization of diverse classes of materials and devices using electric currents, leading to novel spintronic memory and computing approaches. In parallel to spin transfer torques (STT), which have emerged as a leading non-volatile memory technologie, SOT broaden the scope of current-induced magnetic switching to applications that run close to the clock speed of the central processing unit and unconventional computing architectures. In this paper, we review the fundamental characteristics of SOT and their use to switch magnetic tunnel junction (MTJ) devices, the elementary unit of the magnetoresistive random access memory (MRAM). In the first part, we illustrate the physical mechanisms that drive the SOT and magnetization reversal in nanoscale structures. In the second part, we focus on the SOT-MTJ cell. We discuss the anatomy of the MTJ in terms of materials and stack development, summarize the figures of merit for SOT switching, review the field-free operation of perpendicularly magnetized MTJs, and present options to combine SOT, STT and voltage-gate assisted switching. In the third part, we consider SOT-MRAMs in the perspective of circuit integration processes, introducing considerations on scaling and performance, as well as macro-design architectures. We thus bridge the fundamental description of SOT-driven magnetization dynamics with an application-oriented perspective, including device and system-level considerations, goals, and challenges.