Most current device applications of ZnO are hampered by the lack of control over the electrical conductivity. As-grown ZnO usually exhibits n-type conductivity, and the cause of this residual doping is heavily debated. We have performed temperature-dependent Hall measurements and material characterization by secondary ion mass spectroscopy on nominally undoped bulk ZnO crystals as well as on material doped with potential candidates for p-conductivity in order to explore the cause of the background doping and to study the impact of possible candidates for p-doping of ZnO. Also, this paper gives an overview about surface conductivity of high-resistivity ZnO bulk material and discusses how this property might impact the difficult search for p-type ZnO. We will demonstrate the effect of a surface conducting channel on homoepitaxial MgZnO layers grown by liquid-phase epitaxy. The detectability of such a surface layer on an epi sample indicates the high structural quality and low background doping of the layer.