Einstein Equations: Physical and Mathematical Aspects of General Relativity : Domoschool 2018 Four Lectures on Asymptotically Flat Riemannian Manifolds Lectures on Linear Stability of Rotating Black Holes The Bianchi Classication of the Three Dimensional Lie Algebras and Homogeneous Cosmologies and the Mixmaster Universe The Physics of LIGO–Virgo Generation of Initial Data for General Relativistic Simulations of Charged Black Holes BMS Symmetries and Holography: An Introductory Overview Relativity of Observer Splitting Formalism and Some Astrophysical Applications Some Remarks on a New Exotic Spacetime for Time Travel by Free Fall Schwarzschild Spacetime Under Generalised Gullstrand–Painlevé Slicing Crystal Spacetimes with Discrete Translational Symmetry Electrogeodesics and Extremal Horizons in Kerr Newman (anti )de Sitter Hearing the Nature of Compact Objects Minisuperspace Quantisation via Conditional Symmetries This book is based on lectures given at the first edition of the Domoschool the International Alpine School in Mathematics and Physics held in Domodossola Italy in July 2018. It is divided into two parts. Part I consists of four sets of lecture notes. These are extended versions of lectures given at the Domoschool written by well known experts in mathematics and physics related to General Relativity. Part II collects talks by selected participants focusing on research related to General Relativity