A process for hydrogenating olefins is disclosed. The olefins are present in a feed gas which includes H2 and one or more sulfur compounds. The sulfur compounds may include H2S and organic sulfur compounds. The feed gas is passed through a reactor at an inlet temperature from 100° C. to 250° C. The reactor contains a catalyst which is active at the inlet temperature. The reactor may be adiabatic. Saturated hydrocarbons are formed from the olefins. A temperature gradient may be formed in the reactor due to the exothermic nature of the hydrogenation reaction, causing the temperature to increase downstream in the reactor. At temperatures higher than the inlet temperature, H2S may be formed from organic sulfur compounds. A gas mixture including saturated hydrocarbons, H2S and H2 exits the reactor and may be brought into contact with a chemical adsorbent which removes the H2S. The gas stream may then be passed to a steam methane reformer.