Wash apples thoroughly and peel. Cut in quarters, remove and discard core, stem, and blossom ends. Chop apples coarsely.
Place chopped apples in a saucepan with a heavy bottom, and add a small amount of water, just enough to prevent burning.
Heat over medium-high heat until water begins to bubble, reduce heat to low, cover pan, and cook, stirring occasionally, until apple pulp is very soft.
Remove from heat. For chunky applesauce, serve as is, or add sugar to taste.
For smooth applesauce, puree with a food processor or force through a sieve. Serve as is or add sugar to taste.
* Note: To cook applesauce in the microwave, prepare apples as described above. Place in a microwave-safe container, with a few tablespoons of water.Cover and cook at High power, stirring occasionally, until very soft.