Real World article
(written from a Production point of view)

With corporate mergers, acquisitions, and splits, the Star Trek franchise has had several different owners in its history.

Desilu (1964 – 1967)
The studio produced the two Trek pilots "The Cage" and "Where No Man Has Gone Before" in 1964 and 1965 respectively, and began to produce Star Trek: The Original Series in 1966.
Paramount Television, Gulf+Western (1967 – 1989)
Gulf and Western Industries, more commonly known as Gulf+Western, acquired Desilu in 1967 and named it Paramount Television. Gulf+Western had acquired Paramount Pictures in 1966. The original series shared a joint Paramount Television / Desilu copyright in the latter half of the second season, and went under the Paramount Television logo and copyright in the third season.
In 19731974, Paramount Television licensed Filmation to produce Star Trek: The Animated Series, but asserted ownership. Paramount Pictures began producing and distributing Trek films in 1978. Paramount Television began producing Star Trek: The Next Generation in 1987.
Paramount Television, Paramount Communications (1989 – 1994)
Gulf+Western reorganized in 1989, sold off most of its non-communications holdings, and renamed itself Paramount Communications. It continued to hold Paramount Television, which continued to produce TNG and began producing Star Trek: Deep Space Nine in 1993, and Paramount Pictures, which continued to produce Star Trek films.
Paramount Television, Paramount Communications, Viacom (1994 – 2005)
Paramount Communications was purchased in 1994 by the old Viacom, a subsidiary of National Amusements. During Viacom's ownership, Star Trek: Voyager and Star Trek: Enterprise were produced, while Deep Space Nine continued to be produced until is conclusion in 1999. Paramount Pictures continued to produce Trek movies.
TV – CBS Television Distribution, CBS Paramount Network Television/CBS Television Studios, CBS Corporation (2005 – present)
Viacom had purchased CBS Corporation in 2000. They split in 2005. The old Viacom then became CBS Corporation. Its holding Paramount Television became CBS Paramount Network Television. CBS Television Distribution formed soon after and took over distribution of past Star Trek shows. In 2009, CBS Paramount Network Television became CBS Television Studios. Star Trek: Discovery is produced under this ownership.
Movies, DVDs – Paramount Pictures, Viacom (new) (2005 – present)
In the 2005 Viacom/CBS split, the old Viacom became the CBS Corporation and a new Viacom was created. This new company owns Paramount Pictures, which in turn owns the Trek films. Paramount Pictures produced Star Trek, Star Trek Into Darkness, and Star Trek Beyond under license from CBS Television Studios. Paramount also continues to distribute DVDs of the TV series on behalf of CBS. The split marked the occasion that the former Paramount Television was formally separated from Paramount Pictures; until this point in time the television company had always been a subsidiary division of Paramount Pictures.