Recovery
The machine will not start.
Work from the outside in: what is plugged in, then the boot menu, then the system. Most cases stop at the first two, and the ones that do not have an order of their own.
Before anything, take out what is plugged in
A USB stick left in the port is the single most common cause of a machine that "will not start". It starts perfectly, from the stick, and shows you the installer again.
Unplug every USB drive and memory card, and try once more.
Nothing at all on the screen
Then you are not at the system yet. Check the power, the cable, the screen and which input it is showing. On a laptop, hold the power button down for ten seconds, let go, and start it again.
A machine that shows nothing, makes no sound and lights nothing is a hardware question, and no amount of work on the system will move it.
The boot screen appears, then it stops
Hold Shift while the machine starts, or tap Escape, to reach the boot menu. It lists the kernels installed on the machine and an entry for recovery.
Start the previous kernel. If the machine comes up, the newest one is the problem, and you now have a working machine from which to deal with it rather than a dead one.
It starts, but the login screen never comes
From the same boot menu, take the recovery entry. It opens a small menu, in text, with an option to repair broken packages and one to open a root shell.
Repairing broken packages settles the most common case on its own: an update interrupted halfway through, by a power cut or by a machine closed too early.
Undo the last change
If snapshots have been set up on this machine, this is exactly what they exist for. Restore the one dated before the trouble started.
Use the installation stick as a rescue disk
The image runs the full desktop before installing anything. Start the machine from the stick, choose to try the system rather than install it, and you have a working computer with a file manager and a network.
That is enough to copy your files onto an external disk before anything else is attempted. Copy first, repair after. That order costs nothing and it has saved a great many people from losing both.
The disk is encrypted and the passphrase is refused
There is no way in without it. We hold no copy and no recovery key, and that is the entire point of encrypting a disk: a passphrase we could recover would be a passphrase anyone could recover.
Before concluding the passphrase is wrong, check the keyboard you are typing it on. The characters produced by the same keys change with the layout, and a passphrase with an accented letter or a symbol in it is where that bites.
If none of it works
Post on the forum with the exact message on the screen, the machine, and what changed just before it started. Those three lines are the difference between an answer the same day and a thread nobody can help with.
On a paid edition, write to support@cyclop-tech.com: a first answer comes within two working days.