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I Prologue; 1 Introduction; 2 The Initial Value Problem; 3 The Topology of the Universe; 4 Notions of Proximity; 5 Observational Support; 6 Concluding Remarks; II Introductory Material; 7 Main Results; 8 Outline, General Theory; 9 Outline, Main Results; 10 References and Outlook; III Background and Basic Constructions; 11 Basic Analysis Estimates; 12 Linear Algebra; 13 Coordinates; IV Function Spaces, Estimates; 14 Function Spaces, Distribution Functions; 15 Function Spaces on Manifolds; 16 Main Weighted Estimate; 17 Concepts of Convergence; V Local Theory; 18 Uniqueness; 19 Local Existence; 20 Stability; VI The Cauchy Problem in General Relativity; 21 The Vlasov Equation; 22 The Initial Value Problem; 23 Existence of an MGHD; 24 Cauchy Stability; VII Spatial Homogeneity; 25 Spatially Homogeneous Metrics; 26 Criteria Ensuring Global Existence; 27 A Positive Non-Degenerate Minimum; 28 Approximating Fluids; VIII Future Global Non-Linear Stability; 29 Background Material; 30 Estimates for the Vlasov Matter; 31 Global Existence; 32 Asymptotics; 33 Proof of the Stability Results; 34 Models with Arbitrary Spatial Topology; IX Appendices; A Pathologies; B Quotients and Universal Covering Spaces; C Spatially Homogeneous and Isotropic Metrics; D Auxiliary Computations in Low Regularity; E Curvature, Left Invariant Metrics; F Comments, Einstein-Boltzmann

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