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Once eggs are cooked, they should be used within one week. It doesn’t matter if they are peeled or not. If is preferable not to peel them until you are ready to eat or use in your recipe. Peeled hard boiled eggs can be stored in the refrigerator in a bowl of cold water to cover for about 1 week (change the water daily) - or in a sealed container without water (cover the eggs with damp paper towels) for the same length of time.
SAFETY NOTE: It is not safe to leave hard boiled eggs (including those in their shells) out at room temperature for long. If they have been taken to a picnic, or served on a buffet, keep them cool while they are being served, and discard the leftovers.