While it seemed like the Star Trek movie series directed by J.J. Abrams was a success, it clearly didn't do well enough for Paramount to fasttrack more sequels, as we've been waiting years for any sliver of information relating to the fourth film in the series. That being said, Paramount has ordered a new Star Trek film, but unfortunately this won't continue the adventures of Chris Pine's Captain Kirk or see the U.S.S. Enterprise zooming off into deep space.
Deadline notes that while this film will expand the Star Trek universe that Abrams rebooted with the 2009 film, it will do so in the form of a prequel set decades before that very action film. The plot details for this upcoming project remain under wraps, but it is said that Toby Haynes will direct the film and that Seth Grahame-Smith will write the script.
As for Star Trek 4, which is expected to pick up on the narrative following Star Trek: Beyond after it premiered back in 2016, it is noted that this film is still in "active development", although that's not exactly very comforting as it has been in this same phase for many years at this point.