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Aims. Metis is the first solar coronagraph designed for a space mission and is capable of performing simultaneous imaging of the off-limb solar corona in both visible and UV light. The observations obtained with Metis aboard the Solar Orbiter ESA-NASA observatory will enable us to diagnose, with unprecedented temporal coverage and spatial resolution, the structures and dynamics of the full corona in a square field of view (FoV) of ±2.9° in width, with an inner circular FoV at 1.6°, thus spanning the solar atmosphere from 1.7 R⊙ to about 9 R⊙, owing to the eccentricity of the spacecraft orbit. Due to the uniqueness of the Solar Orbiter mission profile, Metis will be able to observe the solar corona from a close (0.28 AU, at the closest perihelion) vantage point, achieving increasing out-of-ecliptic views with the increase of the orbit inclination over time. Moreover, observations near perihelion, during the phase of lower rotational velocity of the solar surface relative to the spacecraft, allow longer-term studies of the off-limb coronal features, thus finally disentangling their intrinsic evolution from effects due to solar rotation. Methods. Thanks to a novel occultation design and a combination of a UV interference coating of the mirrors and a spectral bandpass filter, Metis images the solar corona simultaneously in the visible light band, between 580 and 640 nm, and in the UV H I Lyman-α line at 121.6 nm. The visible light channel also includes a broadband polarimeter able to observe the linearly polarised component of the K corona. The coronal images in both the UV H I Lyman-α and polarised visible light are obtained at high spatial resolution with a spatial scale down to about 2000 km and 15 000 km at perihelion, in the cases of the visible and UV light, respectively. A temporal resolution down to 1 s can be achieved when observing coronal fluctuations in visible light. Results. The Metis measurements, obtained from different latitudes, will allow for complete characterisation of the main physical parameters and dynamics of the electron and neutral hydrogen/proton plasma components of the corona in the region where the solar wind undergoes the acceleration process and where the onset and initial propagation of coronal mass ejections (CMEs) take place. The near-Sun multi-wavelength coronal imaging performed with Metis, combined with the unique opportunities offered by the Solar Orbiter mission, can effectively address crucial issues of solar physics such as: the origin and heating/acceleration of the fast and slow solar wind streams; the origin, acceleration, and transport of the solar energetic particles; and the transient ejection of coronal mass and its evolution in the inner heliosphere, thus significantly improving our understanding of the region connecting the Sun to the heliosphere and of the processes generating and driving the solar wind and coronal mass ejections. Conclusions. This paper presents the scientific objectives and requirements, the overall optical design of the Metis instrument, the thermo-mechanical design, and the processing and power unit; reports on the results of the campaigns dedicated to integration, alignment, and tests, and to the characterisation of the instrument performance; describes the operation concept, data handling, and software tools; and, finally, the diagnostic techniques to be applied to the data, as well as a brief description of the expected scientific products. The performance of the instrument measured during calibrations ensures that the scientific objectives of Metis can be pursued with success. [ABSTRACT FROM AUTHOR]