We explore the properties of group-II acceptors in GaN by performing hybrid density functional calculations. We find that MgGa gives rise to hole localization in zinc-blende GaN, similar to the behavior in the wurtzite phase. Alternative acceptor impurities, such as Zn and Be, also lead to localized holes in wurtzite GaN, and their ionization energies are larger than that of Mg. All these group-II acceptors also cause large lattice distortions in their neutral charge state, which in turn lead to deep and broad luminescence signals. We explore the consequences of these results for p-type doping.