Jackie Chan's Action Kung-Fu
This is a side-scrolling action game starring Jackie Chan. It's not based on one of his movies, but the plot would fit right in with some of his old-school stuff. Jackie has to travel to the top of a mountain to rescue a princess, and along the way he learns new moves to take on the villains with. It's a pretty fun game once you get the hang of it.
This is the US cartridge of the Nintendo version of the game. Notice the nice "Americanized" picture of Jackie. Also of note is the ad campaign, which for some reason didn't generate huge sales for the game. Kung fu doesn't have black belts -- but at least this game didn't have "wimpy" graphics.
Here's a scan of a pirated version of the game. As with movies, Hong Kong is a center for video game piracy and the game was widely made into other versions, such as the one below.
Hong Kong pirates released a "Super Mario" version of this game called "Kung Fu Mari[o]".