In Chap. 30 we begin in Sect. 30.1 with a discussion of the singularity structure of black hole spacetimes by introducing the notion of a “warped product metric”. This is used in Sect. 30.2 to construct a chart for the “exterior spherically symmetric vacuum Schwarzschild spacetime” describing the spacetime outside the history of an isolated spherically symmetric body such as a stable dark star. In Sect. 30.3 we show how to extend the exterior Schwarzschild metric and, using a chart devised by Painlevé and Gullstrand, define the remarkable properties of the spherically symmetric Schwarzschild event horizon. We also analyse radially interior and exterior timelike Schwarzschild geodesics in this chart. In Sect. 30.4, general timelike and null geodesic equations are analysed for the exterior vacuum geometry in Schwarzschild coordinates. In Sect. 30.5, a number of these geodesics are analysed numerically. It is shown explicitly how to calculate perturbatively the “solar deflection of starlight” that can be observed during a solar eclipse from the Earth. This section concludes with a discussion of the stability of certain timelike and null Schwarzschild geodesics and the self-focussing properties of the “photon sphere”. In Sect. 30.6, an analysis of radial timelike Schwarzschild geodesics concludes with explicit calculations of the proper time it takes for a massive particle to fall into a spherically symmetric solar mass, stellar mass or galactic mass black hole. In Sect. 30.7, a special class of equatorial and spiral timelike Schwarzschild orbits is discussed that may have relevance to astrophysical models attempting to understand the characteristics of observed X-ray emissions in the vicinity of compact stellar objects. Such models provide a perspective on the complex processes that lead to the formation of “accretion disks” and “cosmic jets” found in the astrophysics of non-Newtonian gravitation and plasma physics. Section 30.8 discusses, in some detail, a class of Tolman-Oppenheimer-Volkov models for a static, spherically symmetric star leading to a class of Tolman-Oppenheimer-Volkov equations. In Sect. 30.9 we introduce Kruskal-Szekeres charts. It is clear from the above sections that a viable interpretation of the vacuum Schwarzschild spacetimes relies heavily on their symmetry under transformations generated by spatial rotations and temporal translations. These in turn generate local isometries that facilitate certain physical interpretations based on one’s knowledge of the global properties of Euclidean 2-spheres and Euclidean 2-planes. As we have seen in previous sections that explored Schwarzschild spacetimes, the interpretation of their properties was sometimes obscured by the presence of singularities. The question of how some of these singularities can be removed is addressed in Sect. 30.3 by isometrically embedding the exterior Schwarzschild manifold into an extended Painlevé-Gullstrand manifold. In Sect. 30.9 we adopt a different isometric embedding and, in the process, are led to contemplate new spacetimes with “exotic” physical properties. The construction of a spherically symmetric metric for a manifold that was regular except where a Kretschmann scalar becomes singular was completed by Kruskal and Szekeres, based on earlier work by Eddington and Finkelstein. Our approach uses properties of the principle branch of the LambertW-function since it eliminates the need to exploit some of the implicitly defined transformations that are found in many textbook treatments of this construction. Section 30.10 is devoted to the behaviour of causal Schwarzschild geodesics derived with the aid of a Killing tensor field. In Sect. 7.3 the relevance of a Killing tensor field for finding constants of geodesic motion that are not constructed from Killing vector fields was discussed. In Sect. 30.10 we demonstrate how the equations for causal geodesics for the exterior Schwarzschild metric can be derived from a system of first-order ODEs. Section 30.11 demonstrates that spherically symmetric solutions to the coupled system of Einstein and Maxwell field equations, found by Reissner and Nordström can describe an electrically charged black hole. The properties of its causal geodesics are compared with those of an electrically neutral black hole and the non-geodesic equation of motion for an electrically neutral particle in the Reissner-Nordström geometry is discussed. Furthermore, it is shown that, like the Schwarzschild metric, some Reissner-Nordström domains are non-stationary. Finally, an integrable first-order system of ODEs for all causal geodesics is also obtained with the aid of a Killing tensor field.