The above answers solve the problem, though a couple of amendments for versions of Ubuntu with systemd (15.04+), as follows:

• Take advantage of new -auth guess functionality in x11vnc - which helps!
• Update for systemd (not upstart)

Run the following to install:

sudo apt-get install x11vnc
sudo x11vnc -storepasswd yourVNCpasswordHERE /etc/x11vnc.pass
# for Ubuntu 15.04+
sudo nano /lib/systemd/system/x11vnc.service
# for Ubuntu 16.10+
sudo nano /etc/systemd/system/x11vnc.service

Insert this into the file:

[Unit]
Description="x11vnc"
Requires=display-manager.service
After=display-manager.service

[Service]
ExecStart=/usr/bin/x11vnc -xkb -noxrecord -noxfixes -noxdamage -display :0 -auth guess -rfbauth /etc/x11vnc.pass
ExecStop=/usr/bin/killall x11vnc
Restart=on-failure
Restart-sec=2

[Install]
WantedBy=multi-user.target

Then, start with:

sudo systemctl daemon-reload
sudo systemctl start x11vnc

And ensure the service starts on boot:

sudo systemctl enable x11vnc

Install x11vnc:

sudo apt-get install x11vnc

Create a password for your user:

x11vnc -storepasswd

If you have ssh setup you can use it to start x11vnc assuming you are logged in already, but remember to tell it to use your password file:

x11vnc -usepw

If you are not logged in you will get an error with the explanation:

If NO ONE is logged into an X session yet, but there is a greeter login
program like "gdm", "kdm", "xdm", or "dtlogin" running, you will need
to find and use the raw display manager MIT-MAGIC-COOKIE file.
Some examples for various display managers:

 gdm:     -auth /var/gdm/:0.Xauth
          -auth /var/lib/gdm/:0.Xauth
 kdm:     -auth /var/lib/kdm/A:0-crWk72
          -auth /var/run/xauth/A:0-crWk72
 xdm:     -auth /var/lib/xdm/authdir/authfiles/A:0-XQvaJk
 dtlogin: -auth /var/dt/A:0-UgaaXa

Assuming you are using lightdm for the login you can fix this problem you can start x11vnc with the command:

sudo x11vnc -xkb -noxrecord -noxfixes -noxdamage -display :0 -auth /var/run/lightdm/root/:0 -usepw

I am not sure this is the best idea to run x11vnc as root. Maybe someone could edit with a way to access the login without using sudo.

Once this is running you should be able to connect using a vnc client such as krdc (for KDE). You might want to use GNU Screen to keep x11vnc running without needing the ssh session open

I was able to figure this out using http://ubuntuforums.org/showthread.php?t=2039022.

Here is a sample upstart job you can use to make it run at startup. It needs to be put in /etc/init/x11vnc.conf. (Note that newer versions of Ubuntu use systemd so see the other answer that has a sample systemd config):

# description "start and stop x11vnc"

description "x11vnc"

start on runlevel [2345]
stop on runlevel [^2345]

console log
#chdir /home/
#setuid 1000
#setgid 1000

respawn
respawn limit 20 5

exec x11vnc -xkb -noxrecord -noxfixes -noxdamage -display :0 -auth /var/run/lightdm/root/:0 -usepw

Once you have made this file you can start it by running: sudo start x11vnc You can check the log at: /var/log/upstart/x11vnc.log

Here's how:

1. Install the X11VNC server (or through Ubuntu Software Center -> X11VNC Server)

   sudo apt-get install x11vnc
   

2. Create a VNC password file.

   sudo x11vnc -storepasswd yourVNCpasswordHERE /etc/x11vnc.pass
   

3. Create a job file in the editor nano (or gedit, leafpad etc.).

   sudo nano /etc/init/x11vnc.conf
   

4. Paste this into the file:

   start on login-session-start
   
   script
   
   /usr/bin/x11vnc -xkb -forever -auth /var/run/lightdm/root/:0 -display :0 -rfbauth /etc/x11vnc.pass -rfbport 5900 -bg -o /var/log/x11vnc.log
   
   end script
   

5. Save the file. You created a job for the Upstart event login-session-start.

6. Restart Ubuntu.

That's it! You should now be able to connect with any VNC client even before login.

babelmonks answer as a bash script, copy and save as x11vnc.sh & run with sudo bash /path/to/file/x11vnc.sh (sorry dont have enough rep to post a comment)

#!/bin/bash
#install x11vnc & set password
apt-get install x11vnc -y
x11vnc -storepasswd 123456 /etc/x11vnc.pass
#create config file for  system service
cat > /lib/systemd/system/x11vnc.service <<-EOF
[Unit]
Description="x11vnc"
Requires=display-manager.service
After=display-manager.service

[Service]
 ExecStart=/usr/bin/x11vnc -xkb -noxrecord -noxfixes -noxdamage -display :0 -auth guess -rfbauth /etc/x11vnc.pass
ExecStop=/usr/bin/killall x11vnc

[Install]
WantedBy=multi-user.target
EOF
#restart new services &  enable on boot
systemctl daemon-reload
systemctl start x11vnc
systemctl enable x11vnc

Use my script for easy set up: installvncubuntu1604.sh

Usage:

chmod +x ./installvncubuntu1604.sh; sudo ./installvncubuntu1604.sh

Here is a config works for sddm (for KDE 5). Currently, -auth guess does not work for sddm, the auth file is different every time, so we need something like -auth /var/run/sddm/*. The service looks like,

[Unit]
Description="x11vnc"
After=multi-user.target

[Service]
ExecStart=/bin/sh -c "/usr/bin/x11vnc -xkb -noxrecord -display :0 -auth /var/run/sddm/* -rfbauth /etc/x11vnc.pass"
ExecStop=/usr/bin/killall x11vnc

[Install]
WantedBy=multi-user.target

I have to put it after multi-user.target, if after display-manager.service, x11vnc cannot find auth file, maybe sddm generates auth file later. This is tested on Manjaro 18.

I use my own shell: https://github.com/dvdvideo1234/UbuntuBatches/tree/master/x11VNC

start on login-session-start

script

sudo /usr/bin/x11vnc -xkb -auth /var/run/lightdm/root/:0 -noxrecord -noxfixes -noxdamage -rfbauth /var/srv/x11vnc/x11vnc.pass -forever -bg -rfbport <your_port> -o /var/srv/x11vnc/x11vnc.log

end script

into the file:

/etc/init/x11vnc.conf

It makes it auto-start on boot, even on the logon screen it asks for a password.

Install x11vnc package

#sudo apt-get install x11vnc

Then set the password

#x11vnc -usepw

Then create startup script for x11vnc

#sudo nano /etc/x11vnc.sh

In File:

/usr/bin/x11vnc -bg -forever -shared -reopen -usepw

Then save

#sudo chmod 777 /etc/x11vnc.sh

Then add the script file to Control Center=>Startup Application Then Click Add

> Name-----------x11vnc

> Command-----<map the script file

> Comment------service

This is must be required to add for all user

This will run after user login only

If you are working with Ubuntu 19.04, you can check it out with my answer as I figured it out.

babelmonk's answer almost solves my problem. But I got a problem with display manager ( to deal with login page). I'm using Ubuntu 19.04 in which gdm3 is used. And yet, even with 'smart' -guess parameter from x11vnc, I couldn't connect to my Ubuntu. Thus, I changed it to lightdm which works well.

You can change your display manager in this command:

sudo dpkg-reconfigure gdm3

BTW, if you wanna full steps to configure x11vnc, I wrote a full instruction.