The grand riemann hypothesis

Extended Abstract: These lectures are a continuation of Bombieri's series "The classical Theory of Zeta and L-Functions" (in this volume). Naturally there is some overlap between his and our presentations. My aim is to formulate the Riemann Hypothesis "GRH" in its most general setting and to demonstrate its importance and power as well as to indicate some of the progress that has been made around these conjectures. A particular theme being that a number of the striking applications of the GRH have been proven unconditionally by establishing suitably strong approximations thereof.