In a large bowl, cream together the butter and sugar until smooth. Beat in the eggs, one at a time, mixing well after each. Mix in the vanilla. Combine flour, baking powder, and salt; stir into the sugar mixture. Cover dough and chill for at least one hour.
Preheat the oven to 375 degrees F (190 degrees C). Grease cookie sheets. On a floured surface, roll the dough out to 1/2 inch thick and cut into desired shapes using cookie cutters. Place 2 inches apart on to the prepared baking sheets.
Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.