This text is a considerable amplification and modernization of the authors' earlier Essential Relativity. It brings relativity alive conceptually and emphasizes the foundations and the logical subtleties rather than the mathematics or the detailed experiments. It includes 300 exercises and promotes a visceral understanding and the confidence to tackle any fundamental relativistic problem. Following a critical overview of the whole field, special-relativistic kinematics is presented three dimensionally before the mathematical level gradually rises. Four vectors preceded mechanics, four tensors precede Maxwell theory, and three chapters on cosmology end the text. This book brings the challenge and excitement of modern relativity and cosmology at a rigorous mathematical level within the reach of advanced undergraduates, while containing enough new material to interest lecturers and researchers.