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This book is a considerable amplification and modernisation of the author's earlier Essential Relativity. It aims to bring the challenge and excitement of modern relativity and cosmology at rigorous mathematical level within reach of advanced undergraduates and beginning graduates, while containing enough new material to interest lecturers and researchers. Its basic purpose is to make relativity come alive conceptually. Hence the emphasis on the foundations and the logical subtleties rather than on the mathematics or the detailed experiments per se. Aided by some 300 exercises, it 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 precede mechanics, four-tensors precede Maxwell theory, and two of the eight chapters on general relativity roll by before general tensors are needed. Three 'easy' chapters on cosmology round off the work.