We have measured the shear elastic constants of colloidal crystals using a standing cylindrical resonance technique and the viscosity with a variety of rheometers. The elastic constants are well described by Debye-Huckel theory with a renormalized charge. The crystals melt upon addition of electrolyte and the resulting liquid has a viscosity which is strongly dependent upon the electrostatic interactions between the charged colloids. Extrapolation of the elastic constants into the liquid regime, together with a phenomenological relaxation time, allow us to predict the viscosity of the liquid. We also observe a transition from solid to liquid, "shear melting", upon application of sufficiently high stress. This transition is accompanied by an increase in viscosity in going from solid to liquid