Bananas that require further ripening should be left at room temperature, but away from heat or direct sun. To speed ripening, place them in a loosely closed paper bag. Putting an apple in the bag will further speed the process. Once ripened to your liking, bananas can be held at room temperature for a day or two. Then, you can store them in the refrigerator to slow down ripening; although the skins will turn dark, the fruits will remain perfectly edible. You can keep refrigerated bananas for up to two weeks. But never refrigerate unripe bananas: The exposure to cold interrupts their ripening cycle, and it will not resume even if the fruits are returned to room temperature.
You can salvage an overabundance of overripe bananas by peeling them, wrapping them whole or in chunks in plastic wrap, and freezing them. Eat them frozen (a sweet treat in summer) or thaw them and use in baking, where peak sweetness and "mushiness" are desirable.