  We study the existence of monopole bound states saturating the BPS bound in
N=2 supersymmetric Yang-Mills theories. We describe how the existence of such
bound states relates to the topology of index bundles over the moduli space of
BPS solutions. Using an $L^2$ index theorem, we prove the existence of certain
BPS states predicted by Seiberg and Witten based on their study of the vacuum
structure of N=2 Yang-Mills theories.
