What’s the Difference Between the “System32” and “SysWOW64” Folders in Windows?

On 64-bit versions of Windows, you have two separate Program Files folders. But it doesn’t end there. You also have two separate system directories where DLL libraries and executables are stored: System32 and SysWOW64. Despite the names, System32 is full of 64-bit files and SysWOW64 is full of 32-bit files. So what gives?