Method for separating a multicomponent gas mixture of primary component(s) and secondary component(s) into a product stream enriched in the primary component and one or more additional product streams enriched in the secondary component, comprises: (a) introducing the feed gas mixture into a first membrane separation zone; the permeable adsorbent membrane divides the zone into a feed side and a permeate side; the gas passes through the feed side and a portion of the secondary component is selectively adsorbed and permeates through the membrane; (b) withdrawing from the feed side a first non-permeate stream enriched in the primary component; (c) withdrawing from the permeate side a permeate stream enriched in the second component; (d) introducing at least a portion of the permeate stream as a feed gas into a second membrane separation zone having a permeable adsorbent membrane; the feed gas passes through the feed side and a portion of the secondary component is selectively adsorbed by the material of the membrane and permeates through it; an additional product stream enriched in the secondary component is withdrawn; (e) withdrawing from the feed side of the second zone a second non-permeate stream which is further enriched in the primary component, and combining at least a portion of this stream with the multicomponent gas mixture to provide the feed gas to the first zone; and (f) operating the first and second zones such that the ratio of the recovery of the primary component in the first zone to the one in the second zone is less than 1.0; the recovery of the primary component in any zone is defined as the molar flow rate of the primary component in the non-permeate stream withdrawn from the zone divided by the molar flow rate of the primary component in the feed gas introduced into the zone.    USE -   Staged adsorbent membrane systems in multicomponent gas separations. For separating hydrogen, helium, or methane from mixtures of carbon oxides, water, ammonia, nitrogen, hydrogen sulphide and hydrocarbons.    ADVANTAGE -   The method is efficient for operating staged adsorbent membrane systems with a correct balance between product recovery and product purity at acceptable capital and operating costs.