We use data on deaths in New York City, Madrid, Stockholm, and other world cities as well as in various U.S. states and various countries and regions to estimate a standard epidemiological model of COVID-19. We allow for a time-varying contact rate in order to capture behavioral and policy-induced changes associated with social distancing. We simulate the model forward to consider possible futures for various countries, states, and cities, including the potential impact of herd immunity on re-opening. Our current baselinemortality rate (IFR) is assumed to be 1.0% but we recognize there is substantial uncertainty about this number. Our model fits the death data equally well with alternative mortality rates of 0.5% or 1.2%, so this parameter is unidentified in our data. However, its value matters enormously for the extent to which various places can relax social distancing without spurring a resurgence of deaths.