Throughout the various Star Trek shows, there are a number of times when the Enterprise or another Federation vessel encounters a more advanced civilization. For instance, in The Corbomite Maneuver, Balok's ship seems to be more advanced than the Enterprise. Kirk and Balok arrange for a crewmember from the Enterprise to stay with Balok for a while.
In Star Trek: The Next Generation, in The Nth Degree, they meet the Cytherians, who are clearly advanced far beyond the Federation and there is talk of an exchange of technologies.
We know that the Federation has an understanding of the Romulan cloaking device and in the last episode of Star Trek: Voyager we know Captain Janeway goes back in time for a bootstrap operation to adapt future technology so the Voyager can make it home.
In both cases, though, the technology is either not used again or only used again with the supervision of the species who invented it.
Other than the above examples, in the original Trek timeline, is there any example of Starfleet having access to advanced alien technology that was later regularly used by Starfleet? Or any examples of advanced alien technology that were even used at all by Starfleet?
Addendum: When I wrote this, I thought by specifying "the Federation," that would rule out what Voyager used, since we never really had a chance to see what new technology from the Voyager was actually adapted by the Federation after the ship and crew returned from their odyssey. I find it interesting so many examples come from that series, which kind of supports what I was thinking: The trope "Status Quo Is God" (Warning: TV Tropes link!) does come into play with Trek, at least in terms of existing technology. ST:VOY is an exception due to the nature of the series.
I'm still perusing answers and considered that I should rephrase this to specify that if it was just Voyager or only one ship or Captain, that it didn't count, since the intent was to find what tech the Federation used, but, after several answers included Voyager, I don't think it's fair to make that change now.