This paper describes the innovative optical design of the Metis coronagraph for the Solar Orbiter ESA-NASA mission. Metis is a multi-wavelength, externally occulted telescope for the imaging of the solar corona in both the visible and ultraviolet wavelength ranges. Metis adopts a novel occultation scheme for the solar disk, that we named "inverse external occulter", for reducing the extremely high thermal load on the instrument at the spacecraft perihelion. The core of the Metis optical design is an aplanatic Gregorian telescope common to both the visible and ultraviolet channels. A suitable dichroic beam-splitter, optimized for transmitting a narrow-band in the ultraviolet (121.6 nm, HI Lyman-α) and reflecting a broadband in the visible (580-640 nm) spectral range, is used to separate the two optical paths. Along the visible light optical path, a liquid crystal electro-optical modulator, used for the first time in space, allows making polarimetric measurements.