Review of Differential Geometry: manifold, vectors, dual vectors, tensors, covariant differentiation, geodesic, Lie differentiation, Killing vector, curvature, geodesic deviation, Fermi normal coordinates, differential forms, tetrad formalism, Cartan’s structure equation; Geodesic Congruences: energy conditions, kinematics of deformable medium-expansion, shear, rotation, congruence of timelike geodesics-kinematics, Frobenius’ theorem, Raychaudhuri’s equation, focusing theorem, congruence of null geodesics-kinematics, Frobenius’ theorem, Raychaudhuri’s equation, focusing theorem; Geometry of Hypersurfaces: defining equations, normal vector, induced metric, integration on hypersurface, Gauss-Stokes theorem, tangent tensor fields, intrinsic covariant derivatives, extrinsic curvature, Gauss-Codazzi equations, initial value problem, junction conditions for thin shells, gravitational collapse; Lagrangian and Hamiltonian Formulation of General Relativity: Einstein-Hilbert action, variation of the Hilbert term, variation of the boundary term, variation of the matter action, Einstein equation, Bianchi identities, 3+1 decomposition of the metric, field theory, foliation of the boundary, gravitational Hamiltonian, variation of the Hamiltonian, Hamilton’s equations, Hamiltonian definition of mass and angular momentum, Komar formulae, Bondi-Sachs mass, transfer of mass and angular momentum; Reissner-Nordstrom Black Hole: derivation of the metric-solution of Einstein and Maxwell equations, nature of space-time, geodesic in Reissner-Nordstrom space-time-null and timelike geodesics, motion of charged particle; Rotating Black Holes: formation of rotating black hole, derivation of the Kerr metric, reference frame, event horizon, equation of motion, first integrals, motion in the equatorial plane-null and timelike geodesics, General Properties of Black Holes: asymptotic properties of Minkowski space-time, Penrose-Carter conformal diagram, event horizon, Penrose theorem, optical scalars, focusing theorem, Hawking area theorem, apparent horizon, trapped surface, singularities in general relativity, singularity theorems, cosmic censorship conjecture; Stationary Black Holes: no hair theorem, stationary space-time, static space-time, Penrose process, Killing horizon and properties, surface gravity, black hole uniqueness theorem; Black Hole Thermodynamics: black holes and thermodynamics, mass formula, four laws of black hole thermodynamics, generalized second law, entropy as Noether charge, black hole as a thermodynamic system, Euclidean approach, calculation of the Euclidean action, concical singularity method, statistical mechanics of black hole thermodynamics.