Captain John Harriman - U.S.S. Enterprise-B
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Captain John Harriman (Alan Ruck) took command of the newly-commissioned Excelsior-class U.S.S. Enterprise-B in 2203. As seen in Star Trek Generations, Harriman's shakedown cruise for his new starship, which included the media and special guests Captain James T. Kirk, Montogomery Scott (James Doohan), and Pavel Chekov (Walter Koenig), was ill-prepared to encounter the energy ribbon called the Nexus. The inexperienced Captain Harriman's inaugural outing resulted in the supposed death of Kirk. While the circumstances of how Harriman became Captain of the Enterprise were never mentioned on-screen, Star Trek producer Rick Berman offered the character's intended backstory: Harriman was from a wealthy, politically connected family and they bought him command of the Enterprise-B as a stepping stone to a political career.
Captain Rachel Garrett - U.S.S. Enterprise-C
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The only female Captain of the Enterprise, Rachel Garrett commanded the Ambassador-class U.S.S. Enterprise-C, which was caught in a temporal anomaly in 2344 and met Captain Picard's U.S.S. Enterprise-D in 2366, as seen in the Star Trek: The Next Generation episode "Yesterday's Enterprise". Captain Garrett also perished in that episode. Since this was Captain Garrett's lone canonical appearance, the circumstances of how she became Enterprise Captain aren't known, but the apocryphal Star Trek novels developed Garrett's backstory: She served as a Lieutenant on the U.S.S. Argo and as First Officer on the U.S.S. Carthage before Garrett was made Captain of the U.S.S. Enterprise-C in 2332.
Captain Jean-Luc Picard - U.S.S. Stargazer
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While Captain Jean-Luc Picard is closely associated with the U.S.S. Enterprise-D and E, the Federation flagship was not his first starship command. Rather, Picard was Captain of the U.S.S. Stargazer before he was given the Enterprise. After being transferred "dozens of times" (in his own words), Picard was the helmsman aboard the Stargazer in 2333 when his captain was killed on the bridge. Picard assumed command of the vessel, and Starfleet later assigned him the Stargazer's Captaincy. Captain Picard led the Stargazer for over 20 years, and this experience is what prompted Starfleet to assign Picard command of the Galaxy-class Enterprise-D in 2364. Picard would remain Captain of the Enterprise for 21 years before his promotion to Admiral in 2385.
Captain Benjamin Sisko - U.S.S. Defiant
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The first lead of a Star Trek series who wasn't a Captain, Commander Benjamin Sisko took over the former Cardassian space station Terok Nor, which was rechristened by Starfleet as Deep Space Nine, in 2369. Prior to his DS9 posting, Sisko served on the U.S.S. Livingston, the U.S.S. Okinawa, and he was the First Officer of the U.S.S. Saratoga during the Battle of Wolf 359 against the Borg, where his wife Jennifer (Felicia M. Bell) was killed. After three years as DS9's commander, Sisko was finally promoted to Captain in 2371 and he was given command of the U.S.S. Defiant (NX-74205). The unique starship, which Sisko helped design, was built to fight the Borg, and it served Captain Sisko well during the Dominion War.
Captain Kathryn Janeway - U.S.S. Voyager
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Captain Kathryn Janeway (Kate Mulgrew) took command of the U.S.S. Voyager in 2371 but on her first mission to find a missing Maquis vessel, but both starships were swept in a displacement wave that stranded them in the Delta Quadrant. Janeway courageously led Voyager on a seven-year voyage back to Earth, which earned her a promotion to Vice Admiral in 2379. Prior to her tenure on Voyager, Lieutenant Janeway served as Science Officer on the U.S.S. Al-Batini, and then she transferred to the U.S.S. Billings with the rank of Commander, experiences which primed Janeway to become the Captain of the U.S.S. Voyager.