I remember that when I was around 8 or so, it happened several times in the morning while waking up, that I saw someone entering the bedroom while I was not able to speak or move. It was terrifying. It must have been imagination, since nobody appeared to have noticed anything unusual.
There are (creepy) reports from the middle ages of figures sitting on sleepers' chests, legs, moving towards them etc.
It's clearly a biological phenomenon of some description. I've never really looked into it, despite experiencing it often. Related my general take on it below.
Is there a name for such event?