In this paper, we will focus our attention on growth and properties of low-resistivity, n-type ZnO single crystals (mu(e)similar to 230 cm(2)V-s, n(e) similar to 1 x 10(17)/cm(3), at RT), including study and identification of their characteristic shallow residual donors that are thought to control the high electrical conductivity observed in as-grown, n-type ZnO crystals. We will also direct some of our attention to study of the electrical and optical properties of recently demonstrated, Nitrogen-doped, homoepitaxial p-type ZnO films grown by molecular beam epitaxy (MBE), either on low-resistivity (p) substrates, or on Li-diffused, semi-insulating ZnO substrates. Hall mobilities, carrier concentrations and resistivities of the p-type ZnO films ranged from mu(p) = 1-2 cm(2)/V-s, n(p) = 9 x 10(16)/cm(3) to 1-5 x 10(18)/cm(3), and rho similar to 10-40 ohm-cm, respectively. SIMS profiles have shown that the p-type films, with thickness typically in the range of 1-2 mum, contained large concentrations of N-atoms (as high as 1 x 10(19)/cm(3) in the films) against a background of N in the substrates similar to 1 x 10(17)/cm(3), and Temperature-dependent Hall measurements of activation energies have shown that the binding energy (BE) of N-acceptors in ZnO is similar to146 meV, comparable to the BE of shallow Mg acceptors in GaN (similar to 165 meV); conductivity measurements have shown that at least one of our p-type films is prone to instability and type conversion at cryogenic temperatures and under illumination with room light. MBE growth conditions necessary to achieve p-type ZnO:N films/ZnO are discussed together with the characteristic bandedge photoluminescence (PL) signatures of the p-type layers.