When does the game save?

Seriously, under which circumstances does the game save? I've lost about 5 hours of progress so far because the game simply doesn't seem to have a save button.

Yesterday i got to the Rich People's town and then had to go do something else - when i opened the game again it loaded before the first combat of the entire game.

Now i just played until the unlock of the third character and then specifically used the game's "Close Game" Button because i read somewhere else that that might work. It doesn't, i opened the game again to check and i'm Level 4 and about a dozen quests behind.