QUASARS have been detected at many wavelengths, but often ones that are bright at one wavelength are very faint or undetectable at other wavelengths. It has therefore been impossible to design a single search technique that would identify all quasars, raising the question of how many may have gone unidentified. Here we show that quasars selected from a radio catalogue have a wide range of optical colours, which we interpret as arising from varying amounts of dust along the line of sight. Most of this dust probably lies within the quasar host galaxy. If the radio-quiet quasars that would normally be detected optically contain as much dust as the radio-loud ones (and have gone undetected at other wavelengths), then 80% of them have been missed by optical surveys. These missing quasars could adequately account for the observed X-ray background.