Aimed at scientists and mathematicians, this book offers an up-to-date look at the black hole from a mathematical perspective - providing results on matters such as, the determination of the isometry groups and some specific analysis of geodesic orbits. The book presents detailed information on the global geometry of Kerr spacetime - the relativistic model of the gravitational field of a rotating central mass. O'Neill investigates the concepts of curvature, geodesics, isometrics, totally geodesic sub-manifolds and topological structure, with an emphasis on the physical interpretations of these geometrical invariants. The text uses a limited number of notational systems and presents Kerr spacetime as the arena in which physics and related fields operate, from observational astronomy to quantum gravity theory.