(written from a Production point of view)
CBS Studios (formerly the original Paramount Television, CBS Paramount Television, and CBS Television Studios) is a television production company owned by the ViacomCBS that currently holds the production rights of any future Star Trek television series.
The studio produces Star Trek: Discovery along with its companion and aftershow series, Star Trek: Short Treks, After Trek, and The Ready Room.  On 22 April 2019, production on the new, at the time yet-untitled Picard series began. 
Paramount Pictures became the owner of Star Trek franchise, after it had absorbed Desilu in 1967 and incorporated it in its Paramount Television unit. Under this moniker it finished up on Star Trek: The Original Series and went on to become the producer of the Berman-era television series, Star Trek: The Next Generation, Star Trek: Deep Space Nine, and Star Trek: Voyager. The original Viacom acquired Paramount in 1994, then acquired the original CBS Studios in 2000. CBS' studio operations and Paramount's television arm were consolidated into the CBS Paramount Television unit as part of the merger, and as such were responsible for the production of Star Trek: Enterprise.
The television production company took its current form as the result of the late 2005 decision, formalized on 17 January 2006, to split back up the original Viacom conglomerate into two separate publicly traded companies, CBS Corporation and a new Viacom. CBS Corporation acquired Paramount's hitherto combined television dependency, including all rights and title to the entirety of the Star Trek franchise, while now former franchise owner Paramount Pictures, which became henceforth a purely feature film studio under the ownership of the new Viacom, had to negotiate a separate license in order to produce new Star Trek feature films.
On the same occasion the former combined television subsidiary was split into three separate, now independent (meaning, having their own profitability responsibilities) corporate entities, the distributor CBS Television Distribution, the broadcaster CBS Studios (essentially the original broadcaster, it therefore coming full circle) and the television production company CBS Paramount Television, for the time being retaining the original combination name until rechristened CBS Television Studios in 2009. Discovery became the first Star Trek live-action production in which Paramount had no involvement in whatsoever.
The loss of its television division in the 2005 split has never sit well with Paramount, and the studio subsequently decided to institute a brand new Paramount Television Studios production unit in 2013 as a substitute for the one it lost to CBS. Ironically though, stock market analysts expect this unit to be merged with CBS Television Studios in the wake of the December 2019 reunification of Viacom and CBS into ViacomCBS, an expectation reinforced by – besides the logical and obvious cost saving effects such a merger would entail – the circumstance that Paramount motion picture productions have already been added to the backlog library of an as of 2020 yet to be launched unified streaming service intended as successor for CBS All Access.