The talk gives a survey of the present situation in the microscopic theory of pairing and superconductivity in the high-Tc cuprate superconductors. After discussing the general relevance and possibilities of high Tc superconductivity for applications, relevant questions, such as what is the microscopic pairing mechanism and how is the macroscopic super-current coming about, are addressed. Key experiments are outlined and compared with recent theoretical calculations. New numerical (cluster) techniques are reviewed, which aim at a quantitative understanding of microscopic models such as the Hubbard model, as they relate to the unusual properties of the normal state of the high-Tc cuprates and possibly to the microscopic nature of the superconducting pairing. In the last part we also discuss the state of theories unifying anti-ferromagnetism and superconductivity, such as the SO(5) theory high-Tc superconductivity.