We report on the growth of epitaxial ZnO thin films and ZnO based heterostructures on sapphire substrates by laser molecular beam epitaxy (MBE). We first discuss some recent developments in laser-MBE such as flexible ultra-violet laser beam optics, infrared laser heating systems or the use of atomic oxygen and nitrogen sources, and describe the technical realization of our advanced laser-MBE system. Then we describe the optimization of the deposition parameters for ZnO films such as laser fluence and substrate temperature and the use of buffer layers. The detailed structural characterization by x-ray analysis and transmission electron microscopy shows that epitaxial ZnO thin films with high structural quality can be achieved, as demonstrated by a small out-of-plane and in-plane mosaic spread as well as the absence of rotational domains. We also demonstrate the heteroepitaxial growth of ZnO based multilayers as a prerequisite for spin transport experiments and the realization of spintronic devices. As an example, we show that TiN/Co/ZnO/Ni/Au multilayer stacks can be grown on (0001)-oriented sapphire with good structural quality of all layers and well defined in-plane epitaxial relations. ; Comment: 28 pages, 12 figures, 93 references