A process for removing sulfur compounds from hydrocarbon feedstreams, particularly those boiling in the naphtha range, by contacting the feedstream with an adsorbent comprised of cobalt and one or more Group VI metals selected from molybdenum and tungsten on a refractory support. This invention also relates to a process wherein a naphtha feedstream is first subjected to selective hydrodesulfurization to remove sulfur but not appreciably saturate olefins. A product stream is produced containing mercaptans that are removed by use of the cobalt-containing adsorbents of the present invention.