A text for a one-year course at the graduate level, for students with substantial background in algebra. Eight chapters contain material applicable to varieties of every dimension, and six chapters contain material which is particular to the theory of curves. Material considers irreducible varieties over an algebraically closed field, except in one chapter, which works over a finite field. Coverage includes the extension theorem, maps of affine varieties, complete nonsingular curves, and the Riemann-Roch theory. Intersection theory is not covered. Includes chapter exercises. The author teaches mathematics at Stanford University.