This book contains lecture notes of a series of lectures given at the EMS summer school in 1997. The lectures were given to a mixed audience of young mathematicians and physicists with the aim of introducing them to a new, important and quickly growing branch of mathematics. The approach to non-commutative geometry used in the book is based on spectral triples, which is a non-commutative generalization of a Riemannian manifold with a given spin structure and its associated Dirac operator. The first two chapters treat the commutative situation, which is then generalized in the next chapters. The author first introduces axioms for (real) spectral triples. Topics treated in other chapters include the geometry of the noncommutative torus, the noncommutative integral (the Dixmier trace and the Wodzicki residue), the Moyal quantization, equivalences among different geometries and action functionals. New developments in the theory over the last few years are described briefly in the last chapter. The book will be useful both for mathematicians and physicists willing to learn more about non-commutative geometry.

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