Single-screw technology goes back to Archimedes and before. Modern developments really started in the 1950's with the publication of the basic theory of a melt screw pump by the DuPont team (McKelvey, Gore, Squires and coworkers),Darnell and Moll's solids conveying model(1956), screw pull-out experiments by Maddock(1959) and Tadmor's melting model (1966).The barrier screw was invented by Maillefer(1959), various modifications were described in the patent literature and currently most modern screw designs are of this type.Grooved barrel extruder development started with the work at IKV in Aachen, Germany, and this technology is finding wider acceptability nowdays. Subsequent technological developments include various types of screw mixing sections and high rotational speeds. Modeling has been nearly always based on barrel rotation physics with the help of finite differences or finite elements,except for the recent efforts by Campbell and Spalding based on screw rotation. Recent modeling developments include the discrete element method for solids conveying.Output rates have increased dramatically and melt quality has improved significantly.Modeling with reasonably accurate predictive capabilities still remains a challenge.