In a large bowl, cream together butter and sugars until smooth. Add lemon zest. Beat in eggs and vanilla. In another bowl, combine flour, baking powder and salt. Add to wet ingredients slowly. Cover and chill dough for at least one hour (or overnight).
Preheat oven to 400 degrees F. Roll out dough on floured surface to 1/4 to 1/2 inch thickness. Cut into shapes with any cookie cutter. Place cookies 1 inch apart on ungreased cookie sheets.
Bake 6 to 8 minutes in preheated oven. Cool completely.