The Tonight Show, on NBC, weeknights 11:35PM
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This started well before I was born (which was in 1975). It was originally hosted by Steve Allen (1954-57), then Jack Paar (1957-62), then Johnny Carson (1962-92). Carson was the first one I was really aware of. I mean, I didn't get NBC, so I never got to see the show when he hosted it, but... you couldn't not be aware that there was a late night talk show hosted by the guy, even if you knew little more than that. Except that his sidekick was Ed McMahon. Yeah... I didn't know much more than that. I do wish I could have seen the show, back then.
Jay Leno (1992-2009; 2010-14)
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After Carson, the show was hosted from 1992-2009 by Jay Leno When he took over, I still didn't get NBC, so it was awhile later before I got to see the show (probably in 1995). And... it was okay. Definitely amusing enough, I guess, though ultimately I suppose I preferred to watch Late Show, on CBS.
In 2009, Leno left The Tonight Show, and got a 10pm show (The Jay Leno Show), which I don't recall ever watching. So for one season (or less), The Tonight Show would have another host, but when Leno's new show failed, he returned as host of this show.
Conan O'Brien (2009-10)
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When Leno left to do his own thing, The Tonight Show was taken over by Conan O'Brien, of whom I was a fan from his time hosting Late Night, so I watched the first episode. But since I rarely watch late night talk shows anymore, I didn't see much of his tenure until his final episode (which was fun)...
There was a great deal of public drama going on, since Leno's 10pm show failed, and NBC wanted to move him back to 11:35, bumping Conan to 12:05, which Conan balked at. Battle lines were drawn among the fandom (I didn't pay much attention, but I suppose I would have called myself Team Coco.) But ultimately, Leno went back to hosting The Tonight Show, from 2010-14, and later in 2010, Conan got a show on TBS.
Jimmy Fallon (2014-present)
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In 2014, Jimmy Fallon (who had taken over Late Nite when Conan began hosting this show in 2009) took over as host of The Tonight Show. I don't think I've actually seen the show at all since he began hosting it, just because of how rarely I watch late night talk shows. I really should check it out sometime.