Twist extrusion (TE) is one of the most popular techniques of severe plastic deformation, aiming at imparting to a material a tailored microstructure and the associated property improvement. The article provides a survey of the literature on the mechanics of TE and the effect it has on the structure, texture, and the attendant properties of various materials, including metals and alloys, powder materials, and polymers. Special emphasis is placed on vortex flow during TE and its hitherto unexplored potential for producing micro- and macrostructures that promise superior properties of the materials. In particular, the possibility of creating novel hybrid materials with chiral inner architecture is demonstrated. The survey is concluded by a presentation of examples of practical applications of TE.