I was walking the streets of Madrid when I came across the national library. The only problem was that I wasn't allowed in the part where they had books. The building itself on the outside was amazing. I particularly loved the lights on both sides of the staircase that had lamp shades on them. Quite a nice touch.
Inside they have exhibitions and also the permanent collection which is not of books as a library should but the history of books and printing and things book related. Its a great place to visit and best of all, its gratis!