The Ultraviolet and Visible-light Coronagraph (UVC) is one of the solar remote-sensing instruments proposed for the model payload of the Solar Orbiter mission. The Solar Orbiter is one of the two 'Flexible' missions selected in September 2000 by the European Space Agency (ESA) for the definition study phase. A novel orbital design takes the orbiter as close as 0.21 astronomical units (AU) to the Sun, with heliographic latitudes as high as 38 degrees for observations of the solar polar regions at very high spatial resolution. From this vantage point, the UVC can, at the same time, image the visible and ultraviolet coronal emissions and diagnose, with unprecedented temporal and spatial resolution (down to 1200 km) the full solar corona. The UVC's optical design, presented here, consists of an externally occulted, off-axis Gregorian with multilayer-coated optics. The UVC can obtain monochromatic images in the neutral hydrogen HI Lyman (alpha) , (lambda) 121.6 nm, and single-ionized helium HEII Lyman (alpha) , (lambda) 30.4 nm, lines and measure the polarized brightness (pB) of the visible K-corona. The ultraviolet Lyman (alpha) lines are separated with two multilayer coatings mirror and an extreme-ultraviolet transmission filter. The mirrors' coating optimized for 30.4 nm still has a good reflectivity at 121.6 nm and visible. The optical performances, resulting from ray-tracing calculations, are presented here, along with the expected system response to the coronal signal.