Varieties -- Some algebra -- Irreducible algebraic sets -- Definition of a morphism: I -- Sheaves and affine varieties -- Definition of prevarieties and morphism -- Products and the Hausdorff Axiom -- Dimension -- The fibres of a morphism -- Complete varieties -- Complex varieties -- Preschemes -- Spec (R) -- The category of preschemes -- Varieties are preschemes -- Fields of definition -- Closed subpreschemes -- The functor of points of a prescheme -- Proper morphisms and finite morphisms -- Specialization -- Local Properties of Schemes -- Quasi-coherent modules -- Coherent modules -- Tangent cones -- Non-singularity and differentials -- Étale morphisms -- Uniformizing parameters -- Non-singularity and the UFD property -- Normal varieties and normalization -- Zariski’s Main Theorem -- Flat and smooth morphisms.

"The book under review is a reprint of Mumford's famous Harvard lecture notes, widely used by the few past generations of algebraic geometers. Springer-Verlag has done the mathematical community a service by making these notes available once again.... The informal style and frequency of examples make the book an excellent text." (Mathematical Reviews).