Abstract: A linear machine for blow molding hollow bodies includes: an extruder feeding a head for extruding juxtaposed parisons lying on an imaginary vertical plane parallel to or coincident with the imaginary plane containing the longitudinal axis of the extruder itself; mold and counter-mold assemblies with respective grippers which can be alternately moved along an axis contained in the imaginary vertical plane of the extrusion head itself; and a unit for transferring the containers cyclically formed by the molds onto a single conveyor located laterally to the path of the mold and counter-mold assemblies. The conveyor resides longitudinally under the extruder. The mold and counter-mold assemblies can move horizontally. Components cyclically and linearly raise and lower the extruder and/or the head, preventing the parison products from interfering both with each other and with the molds as they come out in a closed condition due to the cooperation with the extrusion head.