We give necessary conditions for the Conway polynomial of a two-bridge link. We apply our conditions to the Alexander polynomial of a two-bridge knot, and obtain simple proofs of the classical theorems of Murasugi and Hartley. We also give sharp bounds for the coefficients of the Conway and Alexander polynomials of a two-bridge link. These bounds improve and generalize those of Nakanishi and Suketa. The efficiency of these conditions are compared on a large number of knots and links.