Using a new way to represent links, that we call a buttery repre- sentation, we assign to each 3-bridge link diagram a sequence of six integers, collected as a triple (p=n;q=m;s=l), such that p q s 2, 0 < n p, 0 < m q and 0 < l s. For each 3-bridge link there exists an innite num- ber of 3-bridge diagrams, so we dene an order in the set ( p=n;q=m;s=l) and assign to each 3-bridge link L the minimum among all the triples that corre- spond to a 3-buttery of L, and call it the buttery presentation of L. This presentation extends, in a natural way, the well known Schubert classication of 2-bridge links. We obtain necessary and sucient conditions for a triple ( p=n;q=m;s=l) to correspond to a 3-buttery and so, to a 3-bridge link diagram. Given a triple (p=n;q=m;s=l) we give an algorithm to draw a canonical 3-bridge diagram of the associated link. We present formulas for a 3-buttery of the mirror image of a link, for the connected sum of two rational knots and for some important families of 3-bridge links. We present the open question: When do the triples (p=n;q=m;s=l) and (p 0 =n 0 ;q 0 =m 0 ;s 0 =l 0 ) represent the same 3-bridge link?