When I leave home, I leave my computer open, and the screen as well.
When I come back, I usually just press any button on the keyboard the screen gets out of the screensaver.
But within one week, this happened to me twice: I press any button, but the screen stays black. And after a while of trying to get out of the screensaver, my computer just restarts, then shuts down automatically, and I have to relaunch it manually.

What the fux?

No one?

Not really enough information for me to contribute. My laptop sometimes fails to go to sleep, so I have to manually reboot it, which sounds similar to what you're saying.

Does it occur when you're idling a specific program? Maybe there's something unique that was running on both of those days, and that's what is causing the issue.

In any case, due to the restart/shut down thing you described, I would say that the monitor is not your problem, but rather a side-effect to the underlying issue.

Do you have a custom screensaver? What's your OS? Did you install anything new in the past couple weeks?

No custom screensaver, I just set it to go black: it wastes less energy (I think).
OS == ? (Windows 7?)
I downloaded torrents of the Seasons 3, 4, 5 and 6 of LOST. But they all seemed extremely safe.

torrents have nothing to do with it. I don't know of any viruses which install automatically. turn off the blank screensaver and see if it still crashes (just manually turn off your monitor.)