This paper provides a tutorial overview over recent vigorous efforts to develop computing systems based on spin waves instead of charges and voltages. Spin-wave computing can be considered a subfield of spintronics, which uses magnetic excitations for computation and memory applications. The Tutorial combines backgrounds in spin-wave and device physics as well as circuit engineering to create synergies between the physics and electrical engineering communities to advance the field toward practical spin-wave circuits. After an introduction to magnetic interactions and spin-wave physics, the basic aspects of spin-wave computing and individual spin-wave devices are reviewed. The focus is on spin-wave majority gates as they are the most prominently pursued device concept. Subsequently, we discuss the current status and the challenges to combine spin-wave gates and obtain circuits and ultimately computing systems, considering essential aspects such as gate interconnection, logic level restoration, input-output consistency, and fan-out achievement. We argue that spin-wave circuits need to be embedded in conventional complementary metal-oxide-semiconductor (CMOS) circuits to obtain complete functional hybrid computing systems. The state of the art of benchmarking such hybrid spin-wave-CMOS systems is reviewed, and the current challenges to realize such systems are discussed. The benchmark indicates that hybrid spin-wave-CMOS systems promise ultralow-power operation and may ultimately outperform conventional CMOS circuits in terms of the power-delay-area product. Current challenges to achieve this goal include low-power signal restoration in spin-wave circuits as well as efficient spin-wave transducers.