When I upgraded my Ubuntu Desktop system from 16.04 to 17.10 using the Ubuntu Software Updater (meaning 16.04 -> 17.04 -> 17.10) I encountered an issue in which my NetworkManager DNS settings failed to work. After some fiddling about I figured out that the issue was related to NetworkManager not being able to write to resolv.conf via resolvconf. In order to fix this execute the following command:
sudo dpkg-reconfigure resolvconf
Then go through the options by tabbing to Ok here and hitting enter:
Then selecting Yes and hitting enter here:
Then hitting enter on the last screen:
Then reboot your system and you should be resolving domain names once again!