How to umount a busy linux device

We all get there at some point.. we happily write:

dune@balthasar:~$ sudo umount /mnt/extUsb

and all we get is a clueless:

umount: can't umount /mnt/extUsb: Device or resource busy

So, who’s keeping the disk busy? Which process do I have to kill to get an umount around here? The answer is easy, and it relies on the linux tool fuser.