There's the typical back and forth of any teen romance in the 'Avatar: The Last Airbender' series, but yes, Zuko and Mai were dating and had a serious relationship. Do they end up married? In 'The Promise', Mai broke up with Zuko due to his erratic behavior as the Fire Lord.
However, there are several comic books that follow 'The Promise', and in 'Smoke and Shadow', they get back together, igniting fans' hopes for a happily ever after. But their actual marital status is not confirmed in the comic or TV series.