Apparatus for energy-efficient conductive-gap membrane distillation includes a feed-liquid source and a distillation module. The distillation module includes a feed-liquid chamber in fluid communication with the feed-liquid source. The feed-liquid chamber includes a selectively porous material that allows a component of the feed liquid to pass through the selectively porous material and exit the feed-liquid chamber in vapor form but not in liquid form. The distillation module also includes a conductive-gap chamber adjacent to the selectively porous material on an opposite side of the selectively porous material from the feed-liquid chamber; a heat-transfer surface maintained at a lower temperature than the feed liquid in the feed-liquid chamber, wherein the heat-transfer surface is in thermal contact with the conductive-gap chamber; and a thermally conductive material extending across the conductive-gap chamber.