after losing much progress, several times, i found this on another forum.
 
there is no manual save option (srsly WHY THE F NOT?). you must rely on the autosave. 
 
the game does NOT save when you exit to the menu (WHWHHYYYYY)
 
the game autosaves when you enter a mission or a proving ground. but it does NOT autosave when you enter the Bastion (SRSLY WHHHYYYYY). so if you do some stuff, return to the bastion, then exit to the menu, do NOT expect the game to save.
 
if you do some stuff and want to quit, here's 2 reliable options.
 
1. enter the next mission or some unfinished proving ground, wait until the little autosave gear animation in the lower left of your screen stops, then power off. when you restart, you will be in the bastion, and your progress will be saved but it will be like you never went into that last mission.
 
2. enter the aettings menu, screw around with audio settings or something. then exit. the game will save your settings and your progress. 
 
IF YOU DO NOT SEE THE LITTLE SPINNY GEAR IN THE LOWER LEFT OF YOUR SCREEN, DO NOT EXPECT THE GAME TO HAVE SAVED YOUR PROGRESS

@phrali: It always saved when I returned to the Bastion.

I think it saves every time you go out of an area, never had problems with the saving system.

It saves when you leave and enter an area. If you return to the bastion and did some work on the forge, arsenal, changed shrines, looked at vigil, bought crap that would be lost if you turn off the console. (happened to me)

One way to 'fix it' is to start a who knows where mission/next bastion mission or a proving ground mission and quit once that is loaded up. You would return back to the bastion when you start again.

Yeah, it made me real paranoid. So I just always loaded up a new area before quitting.

@Dany said:

One way to 'fix it' is to start a who knows where mission/next bastion mission or a proving ground mission and quit once that is loaded up. You would return back to the bastion when you start again.

Exactly what I did.

I already assumed this well in advance. So thankfully I haven't lost anything... yet.

One thing that I tried several times and never worked is loading a new area and quitting to The Bastion. 

bump because people are apparently still having problems with the horribly implemented autosave system

I played through the intro, got to the bastion, then went on to the next area and then quit. When I opened my game up the next day, there was not continue option. I'm very frustrated to have to play through the intro for a 3rd time! (once was the demo).