memory mapped IO: device registers mapped to virtual memory or physical memory?

Hi -

I know memory mapped IO is about mapping device/controller registers to
memory space in order to reduce the number of processor instructions
(and simplify device driver writing) . My question is: are device
registers mapped to virtual memory or physical memory? I suspect the
former, and I have looked up a number of texts but everyone just seems
to skirt about the issue.

Thanks,

- Olumide