PREHEAT oven to 400 degrees. Line the cookie sheets with parchment paper.
Sift the flour, baking powder and salt together; set aside.
Cream the butter and sugar until smooth and fluffy. Beat in the eggs, one at a time, and mix well after each addition. Add the vanilla. In 1 cup measurements, add the dry ingredients to the creamed mixture; mixing well after each addition. Cover and chill the dough for at least 1 hour.
Roll the dough out on a sugared surface (do not put flour down) to 1/2 inches thick. Cut into shapes and place carefully one inch apart, on the prepared cookie sheet. Bake for 6-8 minutes. This is a wonderful basic sugar cookie recipe.