The U.S. Department of Energy has estimated that over 50 GW of offshore wind power will be required for the United States to generate 20% of its electricity from wind. Developers are actively planning offshore wind farms along the U.S. Atlantic and Gulf coasts and several leases have been signed for offshore sites. These planned projects are in areas that are sometimes struck by hurricanes. We present a method to estimate the catastrophe risk to offshore wind power using simulated hurricanes. Using this method, we estimate the fraction of offshore wind power simultaneously offline and the cumulative damage in a region. In Texas, the most vulnerable region we studied, 10% of offshore wind power could be offline simultaneously because of hurricane damage with a 100-year return period and 6% could be destroyed in any 10-year period. We also estimate the risks to single wind farms in four representative locations; we find the risks are significant but lower than those estimated in previously published results. Much of the hurricane risk to offshore wind turbines can be mitigated by designing turbines for higher maximum wind speeds, ensuring that turbine nacelles can turn quickly to track the wind direction even when grid power is lost, and building in areas with lower risk.