If you immediately thought "must use a glitch," you'd be right: this run of Super Mario Bros. 3 posted by TASVideosChannel makes use of a bizarre pipe glitch which allows Mario to tunnel down to princess Peach in world 7-1.

The run was performed by Lord Tom and Tompa, and they describe how they did it here—worth a read, if you're interested in the technical side of all of this. For the rest of us, the idea that Peach is located under an entire glitchworld is kind of funny.

From the video's YouTube description:

Super Mario Bros 3 is a very famous game for the NES. There are normally two main ways to beat it: the quick way using warp whistles to skip worlds (http://tasvideos.org/1590M.html) , and the longer way which goes through all the worlds (http://tasvideos.org/1558M.html) . However, thanks to some esoteric memory corruption involving glitched pipe travel, it's possible to jump right into the princess's room in under three minutes! Bowser is never seen in this run at all.

This run gets to World 7-1 as quickly as possible to perform the glitch, and some unusual moves are done with enemy spawns and positions to get the glitch to work. See the authors' comments (http://tasvideos.org/4288S.html) for some more details. There are also author subtitles provided with the emulator movie file and primary encode to explain some of the key differences in this run.

Lord Tom is petitioning to make runs that use this glitch be a separate category of run, such that previous records that beat the game at any percentage aren't thrown out.

You can even watch them do this glitch on a console, if you'd like: