(written from a Production point of view)
The Arri Group is a motion picture and television equipment company, founded in 1917 by August Arnold and Robert Richter, based in Munich, Germany. It is currently the largest worldwide supplier of film equipment. Arri basically specializes in portable, light-weight cameras, which made their products popular among filmmakers since the 1960s. Since the lightweight Arriflex cameras weren't soundproof, they were mostly used as "MOS" cameras in Hollywood, used only for scenes which didn't require sound recording, while they were popular with European filmmakers as standard shooting equipment (since most European productions were dubbed in post-production, instead of using dialogue recorded on the set).
Arri cameras and lenses were used during the production of Star Trek: The Original Series, as an addition to the standard Mitchell cameras used for filming the series. Besides using it for scenes which didn't require sound recording (see above), usually Arri equipment was used in scenes which required more flexible, hand-held camerawork (for which the robust Mitchell camera wasn't suited for), such as fight scenes, or special optical effects (like the use of the 9mm fisheye lens in "Metamorphosis"). 
Almost five decades later, Star Trek and Star Trek Into Darkness also utilized Arri equipment, notably the Arriflex 435 ES film camera, in addition to the Panavision cameras and lenses mostly used during principal photography.
Star Trek credits Edit
(This list is currently incomplete.)