The 2nd edition of this textbook features more than 100 pages of new material, including four new chapters, as well as an improved discussion of differential geometry concepts and their applications. The textbook aims to provide a comprehensive geometric description of Special and General Relativity, starting from basic Euclidean geometry to more advanced non-Euclidean geometry and differential geometry. Readers will learn about the Schwarzschild metric, the relativistic trajectory of planets, the deflection of light, the black holes, and the cosmological solutions like de Sitter, Friedman-Lemaître-Robertson-Walker, and Gödel ones, as well as the implications of each of them for the observed physical world. In addition, the book provides step-by-step solutions to problems and exercises, making it an ideal introduction for undergraduate students and readers looking to gain a better understanding of Special and General Relativity. In this new edition, a wide discussion on metric-affine theories of gravity and equivalent formulations of General Relativity is reported. The aim is presenting also topics which could be useful for PhD students and researchers studying General Relativity from an advanced point of view.