The viscosity and creep mechanism of Earth's inner core are evaluated based on microphysical models of the flow properties of iron under high pressure and temperature, low stress and large grain size. Harper-Dorn creep, a Newtonian-viscous dislocation mechanism, is shown to be the likely deformation process, and the viscosity is predicted to be ~1011 Pa s, at the low end of previous estimates. Such a low viscosity implies that the inner core can adjust its shape to maintain alignment with the gravitational field imposed by the mantle on a timescale of approximately one minute. It also implies that strain sufficient to produce significant lattice preferred orientation could develop in a few years to a few hundred years, which suggests that seismic anisotropy of the inner core is the product of active deformation and has no memory of primary crystallization.