Abstract: I will describe some applications of Floer homology (holomorphic disks or a conjecturally equivalent theory constructed with Seiberg-Witten monopoles) to certain questions in knot theory. Applications include new bounds on unknotting numbers of knots, and also a Dehn surgery characterization of the unknot. I will describe results obtained jointly with Peter Kronheimer, Tomasz Mrowka, and Zoltán Szabó.