This book, Complex Geometry and Mathematical Physics: Classical and Quantum Singularities of Space-Times (Book III-B), presents results from the theory of singularities of space–times, embracing both classical (curvature singularities, bundle boundary constructions) and quantum aspects (G.T. Horowitz and D. Marolf’s theory of quantum probes of spacetime singularities, resolution of negative mass naked singularities, resolution of singularities from the loop perspective). The second in a captivating series of three books, it addresses the mathematical analysis and the geometry of black holes, for example R. Penrose’s treatment of gravitational collapse, stability of black holes exteriors, wave functions of black holes and B-Y. Chen’s approach to marginally trapped surfaces relying on the geometry of the second fundamental form of an immersion into a Lorentzian manifold. The other two books of the series are:Complex Geometry and Mathematical Physics: Lorentzian Geometry and Field Equations (Book III-A)Complex Geometry and Mathematical Physics: Complex Analysis versus General Relativity Theory (Book III-C)"Complex Geometry and Mathematical Physics" is part of the ampler book project "Differential Geometry, Partial Differential Equations and Mathematical Physics", by the same authors, and aims to demonstrate the interaction between complex analysis and complex geometry on one hand, and general relativity and (quantum) gravity theory on the other, with an emphasis on the modern and contemporary trends of applying ideas from GRG theory to certainproblems arising in complex analysis, such as the many pathologies of the Diederich–Fornæss worm domains.