The neutron star crust, the 1 kilometer region on the surface of a neutron star composed of nuclei and superfluid neutrons, is the physical location of the majority of the observable phenomena. In this talk, I will demonstrate how the equation of state and the composition of a cold neutron star crust depends critically on the symmetry energy below the saturation density and the equation of state of low-density neutron matter. When a neutron star is accreting matter from a companion, the nuclei in the crust may not have time to reach equilibrium. The variation of properties of an accreted crust with different symmetry energies will also described. Finally, I will examine the applicability of the single-nucleus approximation at finite temperature.