The discovery of high temperature superconductivity (HTSC) in oxide compounds has confronted materials scientists with many challenging problems. These include the preparation of ceramic samples with critical current density of about 106 A/cm2 at 77 K and sufficient mechanical strength for large-scale electrotechnical and magnetic applications and the preparation of epitaxial thin films of high structural perfection for electronic devices.The main interest in the growth of single crystals is for the study of physical phenomena, which will help achieve a theoretical understanding of HTSC. Theorists still do not agree on the fundamental mechanisms of HTSC, and there is a need for good data on relatively defect-free materials in order to test the many models. In addition, the study of the role of defects like twins, grain boundaries, and dislocations in single crystals is important for understanding such parameters as the critical current density. The study of HTSC with single crystals is also expected to be helpful for finding optimum materials for the various applications and hopefully achieving higher values of the superconducting transition temperature Tc than the current maximum of about 125 K. It seems unlikely at present that single crystals will be used in commercial devices, but this possibility cannot be ruled out as crystal size and quality improve.