I learned this trick from Julie Dannebaum's Fast and Fresh cookbook from the 1980s. It works perfectly 98% of the time.
When boiling the eggs, add lots of salt to the boiling water - like tablespoons of it. I usually add 1-2 tablespoons of salt in enough water to cook a dozen eggs. For more or less eggs, adjust accordingly, erring on the side of too much versus too little.
After you've cooked the eggs (about 12 minutes for large eggs) and cooled them quickly in very cold water (about 5 minutes), they should peel easily and without damage to the egg white.
Contributed by Debbie Heller