In a large bowl, cream butter and sugar until light and fluffy. Add eggs, one at a time, beating well after each addition. Beat in vanilla. Combine the flour, baking powder and salt; gradually add to the creamed mixture (the dough will be soft).
Drop by teaspoonfuls 2 in. apart onto greased baking sheets. Bake at 375° for 7-8 minutes or until the edges are golden brown. Remove to wire racks to cool.