Advances in Geometry and Lie Algebras from Supergravity This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry supergravity. The contents are mainly of mathematical nature but each topic is introduced by historical information and enriched with motivations from high energy physics which help the reader in getting a deeper comprehension of the subject. . 1 Finite Groups and Lie Algebras 2 Isometries and the Geometry of COSET Manifolds 3 Complex and Quaternionic Geometry 4 Special Geometries 5 Solvable Algebras and the TITS Satake Projection 6 Black Holes and Nilpotent Orbits 7 E7 F4 and supergravitiy scalar potentials 8 (Hyper)Kaehler Quotients Ale Manifolds and Cn Γ Singularities 9 Epilogue 10 The Bibliography.