Are 32 bit apps compatible with 64 bit Windows?

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If I upgrade from a 32 bit Windows XP platform to a 64 bit Vista or Windows 7, will all of my 32 bit programs run on it? Or will I be stuck having to upgrade all of my programs to 64 bit versions? My assumption is that some will and some won't. If so, is there a compatibility list somewhere that I can determine what will need upgrading or replacement?
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