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Context. The Extreme Ultraviolet Imager (EUI) is part of the remote sensing instrument package of the ESA/NASA Solar Orbiter mission that will explore the inner heliosphere and observe the Sun from vantage points close to the Sun and out of the ecliptic. Solar Orbiter will advance the "connection science" between solar activity and the heliosphere. Aims. With EUI we aim to improve our understanding of the structure and dynamics of the solar atmosphere, globally as well as at high resolution, and from high solar latitude perspectives. Methods. The EUI consists of three telescopes, the Full Sun Imager and two High Resolution Imagers, which are optimised to image in Lyman-α and EUV (17.4 nm, 30.4 nm) to provide a coverage from chromosphere up to corona. The EUI is designed to cope with the strong constraints imposed by the Solar Orbiter mission characteristics. Limited telemetry availability is compensated by state-of-the-art image compression, onboard image processing, and event selection. The imposed power limitations and potentially harsh radiation environment necessitate the use of novel CMOS sensors. As the unobstructed field of view of the telescopes needs to protrude through the spacecraft's heat shield, the apertures have been kept as small as possible, without compromising optical performance. This led to a systematic effort to optimise the throughput of every optical element and the reduction of noise levels in the sensor. Results. In this paper we review the design of the two elements of the EUI instrument: the Optical Bench System and the Common Electronic Box. Particular attention is also given to the onboard software, the intended operations, the ground software, and the foreseen data products. Conclusions. The EUI will bring unique science opportunities thanks to its specific design, its viewpoint, and to the planned synergies with the other Solar Orbiter instruments. In particular, we highlight science opportunities brought by the out-of-ecliptic vantage point of the solar poles, the high-resolution imaging of the high chromosphere and corona, and the connection to the outer corona as observed by coronagraphs. [ABSTRACT FROM AUTHOR]