DSi Ware games and other downloadable content
The Nintendo DSi Shop is specific to the region for which the system was manufactured. For example, a Japanese system can download DSi Ware games and other software only from the Nintendo DSi Shop for the Japanese region, regardless of where the system is being played.
Nintendo DSi exclusive game cards
Nintendo DSi game cards that are only compatible with Nintendo DSi and Nintendo DSi XL (and not Nintendo DS or Nintendo DS Lite) are region locked. This means game cards manufactured for the U.S. market would not work in a system made for Another Country.
Nintendo DS / Nintendo DS Lite game cards
Older Nintendo DS and DS Lite software is region-free, so you can play most of that software on a Nintendo DSi and Nintendo DSi XL from any region. It's possible, however, that future Nintendo DS software will incorporate region encoding. In these instances, these games would not work in a system made for another country.
Source: http://www.nintendo.com/consumer/systems/dsi/en_na/faq.jsp#vers
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No
Yes, the original Nintendo DS as well as the Nintendo DS lite can play NDSi games.
The Nintendo 3DS will be able to play Nintendo DS and DSi games but games published for the Nintendo 3DS will not be compatible with those systems.
Yes, because the Nintendo DSi is not region locked
Yes, the Nintendo DSi can play Nintendo DS games. The NDS and NDSi are compatible with each other and can link for multiplayer as well. There are really no such thing as 'DSi games' except for DSi-ware and one or two games that require the DSi camera, none of which are compatible with a DS. However, since the DSi has no GBA slot, Game Boy Advance games and DS games that require this slot (e.g. Guitar Hero: On Tour) cannot be played on a DSi. Flash carts and unofficial hardware won't work either, but there may be a newer version available, designed for the DSi. Another problem you might find is that if you have any Wi-fi capable games, they will only work with a WEP security enabled network (or of course a network without security), not WPA.