I have a gas stove, and I'll leave it to simmer, well its really more like boiling, a slow boil/hot simmer, all night. I cover it, make sure there is nothing near the stove to catch fire, and go to sleep fine. I do make sure there is a LOT of liquid, so that it won't accidentally boil dry. I usually add more water than I really need, and let it boil down in the morning. I used to freak out about it, but then I realized, the stove isn't going to explode any more than it would cooking it while I was awake, and its really rather unlikely anything would happen. I do agree, if I had a cat I'd not do it, or shut the cat out of the kitchen. Perhaps not with young kids.... depends on the kids.
I have left it on my stove, with the stove turned off, directly after boiling, for up to 8+ hours. I find that between the pilot light and residual heat, it stays very hot. Its probably a slight gamble, but I don't think its huge. I have no qualms about turning it off for a couple of hours if I go out. But I sometimes leave it on if I go out briefly as well.