For the most part, yes, they will run just fine. I have not experienced any issues. Before posting this answer, I looked on the Microsoft website and they had this to say:
"Applications which most commonly have compatibility issues are what are referred to as kernal mode applications and old 16-bit applications. Examples might include old versions of DOS-based applications, anti-virus applications, old hardware which may not support 64-bit".
So can I run 32 bit apps in Windows 7 64bit? 
