Promptly remove hard-cooked eggs from the cooking and cooling water. Leaving them sitting underwater can foster bacteria growth. (If you peel any hard-cooked egg and the white feels slimy, it's a sign that bacteria have begun to grow, and the egg should be discarded.)
If you've cooked eggs with a week's worth of lunches in mind, leave them in the shell. "The shell is the best form of protection a hard-cooked egg has," says American Egg Board spokeswoman Linda Braun. Stored dry and refrigerated, the eggs will keep for about 1 week. Once peeled, they should be used immediately.
Keep in mind that, like their raw counterparts, hard-cooked eggs can absorb flavors and odors from foods on neighboring refrigerator shelves. It's best to keep smelly foods like onions and cheese in airtight containers.