This page is dedicated to the listing of exploits used used to run homebrew on the Nintendo DSi. Anyone may contribute to this list, as long as any exploits added are explained and verifiable.
Team Twiizers have found a DSi-Mode Exploit and have managed to use it to run DSi Mode homebrew. However it has not yet been released. More details at :  The additional hardware is just required to get a connection to a computer so that things like ram dumps can be created.
Wintermute has made available an open source DSi hack. The exploit works on DSi enhanced games, allowing you to run custom code from a save file. Instructions for using the exploit can be found here: 
This type of exploit is undesirable because all DSi functionality, such as usage of the cameras, is unavailable to homebrew.