Depends on what version of the characters we're talking about here.
Movies: Rogue and Gambit haven't even met T^T She's stuck with Bobby.
Ultimate comics: Gambit kidnaps Rogue and eventually they end up liking each other.
comics: It stared when Gambit started flirting with Rogue in one of the comics. The writers said that originally it wasn't supposed to amount to anything, but the fans liked them. They became a couple :D
Evolution: Gambit kidnapped her, got her to help him and that was the end of it. Although you could tell that Rogue sort of liked him by the end of it. When she is being controlled by Apocalypse in a later episode, she absorbs Gambit's powers by kissing him.
X-Men the Animated Series: They were really flirty through out the series and eventually ended up together.
Wolverine and the X-Men: They haven't met :'( BUT Rogue doesn't have a romantic interest yet, so hopefully they meet next season :D
If you want more info there's some on Rogue's wiki page in her bio. Here's the link:
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