You can install a package resolvconf, which will modify the way /etc/resolv.conf is built up at system boot.

sudo apt install resolvconf

You can then create or modify a file /etc/resolvconf/resolv.conf.d/tail. If you put in this file a line nameserver 8.8.8.8, this line will be added at the end of /run/resolvconf/resolv.conf at boot. /etc/resolv.conf will now be a symbolic link to this file.

Post Scriptum:

Almost two years after posting my answer I came across https://bugs.launchpad.net/ubuntu/+source/ppp/+bug/1778946 which explains exactly why merely installing resolvconf solved a dns problem I had at the time. I feel I have to share this here.

Although my answer addresses the question in a proper way, indeed a warning should be added that, if you want/need to do this, probably something else is wrong. This was already stated by @intelfx at the time, 127.0.0.53 should work by itself.

According that bug report, after a pptp VPN goes down, resolv.conf is restored with the wrong access rights. ping ubuntu.com does not work, sudo ping ubuntu.com does. Installing resolvconf solved it, because it takes over resolv.conf, restoring it with correct rights. Changing systemd-resolve settings is no solution in this case, since the bug is in ppp. But an alternative, maybe simpler solution is sudo chmod a+r /etc/resolv.conf after VPN down. And this can be automated by putting an executable script in /etc/NetworkManager/dispatcher.d with contents:

#!/bin/sh
if [[ "$1"="ppp0" && "$2"="vpn-down" ]]; then  
    /bin/chmod a+r /etc/resolv.conf
fi

In all cases the contents of resolv.conf do not change. And, yes, I know pptp must be avoided because of security issues, but at the time I thought of it as a good excercise for an ubuntu newbie. I imagined it would work out of the box. Little did I know that it would give me a headache, as diagnosed so well by @intelfx.

The correct solution would be to fix systemd-resolved instead of trying to cure migraine with a guillotine.

It is a nice tool, really, if used properly.

Judging by your systemd-resolve --status output...

Link 3 (wlo1)
      Current Scopes: LLMNR/IPv4 LLMNR/IPv6
       LLMNR setting: yes
MulticastDNS setting: no
      DNSSEC setting: no
    DNSSEC supported: no

...your network manager tool does not communicate per-interface DNS configuration to systemd-resolved.

Recent versions of NetworkManager will detect if /etc/resolv.conf is a symlink to a resolved-generated file (/run/systemd/resolve/*.conf or /usr/lib/systemd/resolv.conf) and if that's the case, communicate with resolved directly.

Alternatively, recent versions of systemd-resolved try to be compatible with the historical resolvconf interface by installing a resolvconf binary that does the right thing.

While either of these two solutions would be preferred, if you are looking for a quick and dirty fix, you can just configure systemd-resolved to use your DNS servers globally:

$ cat /etc/systemd/resolved.conf
<...>
[Resolve]
DNS=8.8.8.8 8.8.4.4
<...>

Then restart systemd-resolved.service or reboot.

working within the systemd paradigm add a DNS to a link / device

• systemd.network manual page

using ubuntu 17.10+ add a *.network file:

sudo nano /lib/systemd/network/100-somecustom.network:

100-somecustom.network ( 100 can be any number for priority, and it requires the .network file extension ):

[Match]
Name=wlo1 # the device name here

[Network] # add multiple DNS 
DNS=8.8.8.8
DNS=208.67.222.222

Then restart:

sudo service systemd-networkd restart

Also look into:

netplan apply

Then check:

systemd-resolve --status wlo1

From info page info systemd.network :

In addition to /etc/systemd/network, drop-in ".d" directories can be placed in /lib/systemd/network or /run/systemd/network directories. Drop-in files in /etc take precedence over those in /run which in turn take precedence over those in /lib. Drop-in files under any of these directories take precedence over the main netdev file wherever located. (Of course, since /run is temporary and /usr/lib is for vendors, it is unlikely drop-ins should be used in either of those places.)

Another approach disable the DNSStubListener for usage with dnsmasq:

sudo nano /etc/systemd/resolved.conf:

#
DNSStubListener=false

related:

• https://superuser.com/questions/1153203/ubuntu-17-04-systemd-resolved-dns-lookups-randomly-fail
• DNS at systemd's 127.0.0.53 is ignoring some lookups
• https://unix.stackexchange.com/questions/304050/how-to-avoid-conflicts-between-dnsmasq-and-systemd-resolved
• https://moss.sh/name-resolution-issue-systemd-resolved/

I use Lubuntu & Kubuntu 18.04. I was able to overcome the DNS problem in 2 steps:

First step: Install unbound and set it to replace systemd-resolved as Grégoire C shows here.

sudo systemctl disable systemd-resolved
sudo systemctl stop systemd-resolved
sudo systemctl enable unbound-resolvconf
sudo systemctl enable unbound

reboot

Open as root the file /etc/NetworkManager/NetworkManager.conf

(sudo leafpad /etc/NetworkManager/NetworkManager.conf in Lubuntu 18.04 or in Kubuntu 18.04 SUDO_EDITOR=kate sudoedit /etc/NetworkManager/NetworkManager.conf)

and below [main] put this line:

dns=unbound

reboot again

Second step: If after the reboot the problem is still not solved, as it was in my case, start the file manager as root, go to /etc, delete resolv.conf and create a new resolv.conf. Leave it empty and reboot the OS. In my case after this reboot the problem disappeared.

In Kubuntu 18.04 you can't start Dolphin as root, so you can first go to /etc and open the terminal from there, then type sudo su and press Enter to use it as root and then delete resolv.conf with the command rm resolv.conf. Then you can create a new empty file on your desktop, name it resolv.conf and open a terminal from there. Use the sudo su command to enter the root mode for the terminal and then copy the new resolv.conf from your desktop to /etc with the command cp resolv.conf /etc.

I have to add that I did not try to do the second step before installing unbound, so the next time I will do so, just to see if it will be enough or not.

I'm using Ubuntu 18.04 LTS. I don't know how my OS came into this problem in a sudden while my DNS had been working well for months. Here is my solution:

Install resolvconf

$ sudo apt install resolvconf

Find the source of /etc/resolv.conf.

$ ls -al /etc/resolv.conf shows that /etc/resolv.conf is a symbolic link of /run/resolvconf/resolv.conf

And I found that: This file /run/resolvconf/resolv.conf is composed of:

/etc/resolvconf/resolv.conf.d/head
/run/resolvconf/interface/systemd-resolved
/etc/resolvconf/resolv.conf.d/tail

$ cat /run/resolvconf/interface/systemd-resolved shows nameserver 127.0.0.53, which would be reset after reboot.

Add DNS

It seems that nameserver <my DNS> needs to be in front of nameserver 127.0.0.53 so that it could work.

So I add nameserver <my DNS> at the end of /etc/resolvconf/resolv.conf.d/head.

Update resolvconf

$ sudo resolvconf -u

This is how I change my dns record on the interface configuration.

$ vi /etc/netplan/50-cloud-init.yaml

Change name server addresses, previously it was .4 then I changed it to .3:

...
            nameservers:
                addresses:
                - 192.168.1.3
...

Save the configuration and then apply the configuration:

$ sudo netplan apply

After that restart the resolvectl service.

$ sudo systemctl restart systemd-resolved.service

For notes: I did reboot the server, and changes that I made were still intact. I use the resolvectl dns command to verify the dns record.

Had some issues with NordVPN so decided to add to this.

Findings:
NordVPN sets /run/systemd/resolve/resolv.conf DNS settings when connecting.
NordVPN removes all DNS settings from /run/systemd/resolve/resolv.conf when disconnecting.
This causes DNS server to no longer work (since there isn't one set).

The system needs a reboot to set it back to anything at all. A standard restart of the service (sudo systemctl restart systemd-resolved.service) does not work.

The solution to work around this:

sudo apt install resolvconf

Head over to /etc/resolvconf/resolv.conf.d/

cd /etc/resolvconf/resolv.conf.d/
sudo nano tail

Add nameservers

nameserver 1.1.1.1
nameserver 1.0.0.1

Check following places for nameservers that could have been set:

/run/resolvconf/interface/original.resolvconf
/run/resolvconf/interface/systemd-resolved
/etc/resolvconf/resolv.conf.d/tail
/etc/resolvconf/resolv.conf.d/head
/etc/systemd/resolved.conf
/run/systemd/resolve/resolv.conf
/run/systemd/resolve/stub-resolv.conf

Test which DNS servers are being used with:

nslookup google.com

Server:         1.1.1.1
Address:        1.1.1.1#53

Maybe a bit late, but I ran into this problem. I do have to admit that I have been manually configuring my network settings via cli before and not via the inbuilt networkmanager. I run ubuntu 18.04.

So after I disabled and then re-enabled systemd.resolved.service, this erratic behaviour of randomly losing my DNS configuration was solved for me.

systemctl disable systemd.resolved.service
reboot

after reboot:

systemctl enable systemd.resolved.service
reboot

I am a relatively fresh Linux user so there must be someone that can explain the hows and why's of why this would have worked for me, but it seemed worth to mention if it can help others.

In case you want to manage the file /etc/resolv.conf manually "man NetworkManager.conf" provides the answer.

In /etc/NetworkManager/NetworkManager.conf, section [main] add: dns=none

also make sure /etc/resolv.conf is a file, not a symlink. Your changes will survive system reboot.

Most answers are unfortunately wrong or unnecessary workarounds.

I think you just missed something pretty serious.

install nmap and then run nmap 127.0.0.1, then 127.0.0.53 They have different a list of services, something is definitely listening and hiding in another ip trying to look like localhost. Then the computer is pointing to this address rather than the actual dns server... and resolving... but I did not install any dns server...

I'm looking into what that is... localhost subnet is only supposed to have 127.0.0.1 if I remember correctly.

It sounds like a possible hack to get around firewalls. My firewall was on and rejecting incoming and outgoing, but my outgoing should have been to one ip. My actual dns to work...