This edition reproduces the 2nd corrected printing of the 3rd edition of the now classic notes by Professor Serre, long established as one of the standard introductory texts on local algebra. Referring for background notions to Bourbaki's "Commutative Algebra" (English edition Springer-Verlag 1988), the book focusses on the various dimension theories and theorems on mulitplicities of intersections with the Cartan-Eilenberg functor Tor as the central concept. The main results are the decomposition theorems, theorems of Cohen-Seidenberg, the normalisation of rings of polynomials, dimension (in the sense of Krull) and characteristic polynomials (in the sense of Hilbert-Samuel).