I answered this on this SO question but here's that answer, reprinted.

For 14.04 through 20.04 do the following:

1. Download the drivers from:
   https://www.usa.canon.com/internet/portal/us/home/support/details/printers/black-and-white-laser/mf4770n?tab=drivers#Z7_MQH8HIC0L88RB0AMD0F1Q42K25

   
   



2. Extract the file archive to a directory. From that directory:

   
   

   cd ~/Downloads
   
   tar -xf *inux_UFRII*.tar.*
   
   sudo dpkg -i *inux_UFRII*/*/Debian/*.deb
   
   sudo apt-get install -f
   
   

   
   



3. It shouldn't be necessary to reboot, but if the next step fails, reboot and try again.

   
   



4. Run "Add printer". That should just show up automatically when you click the 'Add' button. Give it a few seconds and the printer chirps then just magically shows up.

   
   

EDIT 2/27/16 (16.04beta) Updated the link to the 3.10 driver (they updated their website). Everything just worked using above, didn't need to reboot.

EDIT 9/8/16: The current driver version is now 3.20. Make installation commands agnostic to both the driver version and the system architecture.

EDIT 06/12/20: The current driver version is now 5.10. Adjusted the installation steps for different wildcards.