The method and furnace according to the invention enable a continuous processing of anode slime and are particularly suited to be connected to a process where anode slime is treated by hydrometallurgic methods after roasting. In the method according to the invention, the slime containing valuable metals and selenium is dried, roasted, sulfatized and cooled. The method includes steps to be carried out in succession, in continuous operation, so that the slime forms a slime layer on the conveyor and is conveyed to be treated in successive drying, roasting, sulfatizing and sulfuric acid removal and cooling units.