In a large bowl, cream together the butter and sugar until smooth. Beat in the eggs and vanilla.
Stir in the flour, baking powder, and salt. Cover and chill dough for at least one hour.
Preheat oven to 400 degrees.
Roll out the dough on a floured surface until 1/4-inch thick. Cut into shapes with cookie cutters. Place cookies 1 inch apart on ungreased cookie sheets.
Bake 6 to 8 minutes or until lightly browned. Remove from the oven, and cool on the pan for five minutes. Transfer to a wire rack. Repeat with the remaining dough.