Hard X-ray surveys have uncovered a large population of heavily obscured AGN. They also reveal a population of quasars with moderate obscuration at both visible and X-ray wavelengths. We use Chandra selected samples of quasars from the ELAIS Deep X-ray Survey (EDXS) and CDF-N to investigate the obscuration towards the nuclei of moderately obscured AGN. We find an inverse correlation between the optical to X-ray flux ratio and the X-ray hardness ratio which can be interpreted as due to obscuration at visible and X-ray wavelengths. We present detailed optical and near-infrared data for a sample of optically-faint (R>23) quasars from the EDXS. These are used to constrain the amount of rest-frame UV/optical reddening towards these quasars. It is found that optically-faint quasars are mostly faint due to obscuration, not because they are intrinsically weak. After correcting for reddening, the optical magnitudes of most of these quasars are similar to the brighter quasars at these X-ray fluxes. Combining with gas column densities inferred from the X-ray observations we consider the gas-to-dust ratios of the obscuring matter. We find that the quasars generally have higher gas-to-dust absorption than that seen in the Milky Way - similar to what has been found for nearby Seyfert galaxies. We consider the possible existence of a large population of X-ray sources which have optical properties of Type 1 (unobscured) quasars, but X-ray properties of Type 2 (obscured) quasars. We show that the observed distribution of optical-to-X-ray flux ratios of quasars at z>1 is skewed to low values compared to the intrinsic distribution...
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