An LED spotlight has an arrangement having at least one LED, above which there is arranged a color conversion substance, and an optical system of a substantially transparent material, which directs incident light from the LED arrangement in the direction of an exit surface by means of total internal reflection. The base surface of the conical optical system may thereby have the shape of a polygon. The outer wall of the conical optical system may have at least in partial regions planar facets. The color conversion material may be provided in the shape of a cuboid above the LED arrangement.