Here is a short review of all possible combinations:
Backward compatibility is not preserved in kernel mode: 32-bit drivers won't work on 64-bit OS!
- 32-bit applications can run both on 32-bit and 64-bit operating systems. 64-bit Windows OS uses WoW64 emulation layer to support executing 32-bit applications which are, by default, installed in C:\Program Files (x86) directory, load 32-bit libraries from C:\Windows\SysWOW64 and use HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node portion of the registry.
- 64-bit applications can run only on 64-bit operating systems. It is possible to run 64-bit operating system on (physically) 32-bit machine (by using some of Virtual Machines, e.g. Oracle's VirtualBox).
It is possible to build applications that target 64-bit architectures on 32-bit hosts. This process is called cross-compiling.