For this speed run, you must complete the game as fast as you can, using glitches as necessary. The backwards long jump allows you to pass the endless stairs without 70 stars. It was eventually posted on the GameFAQs board in March 2003, and it was found that the backwards long-jump also allowed you to pass through the 50-star door. In May 2004 Dom Dunc discovered using MIPS (the rabbit) it was also possible to get through the 30-star door. Since MIPS first appears at 15 stars, and "Board Bowser's Sub" is needed before you can face the second Bowser, it is possible to beat the game with a mere 16 stars. (And with tool-assistance, 0 stars.)
Timing follows SDA rules, and starts from when you first gain control of Mario and ends once Bowser's final star is reached. Some previously completed runs are listed here, along with star choices and times.
UPDATE April 3, 2015: It has been discovered that Myles cheated in his 17:31 video. So that the best set of star times I have on this page aren't dishonest, I've added the star times of the current 16-star record by Xiah. The 16-star record has improved considerably in the intervening time, and by SDA timing his run finishes in 14:39, nearly 3 minutes faster than Myles' claimed time! The most notable improvement is using a backwards long jump in the lobby to get into Bowser in the Dark World with 0 stars. This allows one to get into the basement without getting 8 stars, and consequently completely bypass Bob-omb Battlefield and Cool, Cool Mountain. Instead, 3 extra stars are chosen in Lethal Lava Land in addition to Swimming Beast in the Cavern in Hazy Maze Cave.
Also note that there have been 0-star speedruns in the meantime, making this category mostly of historical interest.