Plasma-assisted molecular beam epitaxy of ZnO epilayers and MgZnO/ZnO heterostructures on Al2O3(0001) substrates is described. A thin MgO layer is employed as a buffer for ZnO. The influence of the buffer on the initial growth of ZnO is discussed with the corresponding reflection high-energy electron diffraction (RHEED) studies. We found that the MgO buffer promotes the lateral growth of ZnO, which results in two-dimensional growth. A 3 X 3 reconstruction is observed and the RHEED intensity oscillations are recorded. The RHEED oscillations have been used in situ to monitor and control the growth of MgZnO/ZnO heterostructures. MgZnO/ZnO single-quantum-well structures have been grown and studied by cathodoluminescence.