We prove that when the equations are restricted to the principal part the standard version of the BSSN formulation of the Einstein equations is equivalent to the NOR formulation for any gauge, and that the KST formulation is equivalent to NOR for a variety of gauges. We review a family of elliptic gauge conditions and the implicit parabolic and hyperbolic drivers that can be derived from them, and show how to make them symmetry-seeking. We investigate the hyperbolicity of ADM, NOR and BSSN with implicit hyperbolic lapse and shift drivers. We show that BSSN with the coordinate drivers used in recent "moving puncture" binary black hole evolutions is ill-posed at large shifts, and suggest how to make it strongly hyperbolic for arbitrary shifts. For ADM, NOR and BSSN with elliptic and parabolic gauge conditions, which cannot be hyperbolic, we investigate a necessary condition for well-posedness of the initial-value problem.