This volume offers a nearly self-contained introduction to some of the basic concepts of algebraic geometry. Prerequisites have been kept to a minimum in order to examine the following areas and some of their standard applications: Bézout's Theorem, the Fundamental Theorem of Projective Geometry, and Zariski's Main Theorem. The exposition is modern, but in the language of “varieties”, rather than that of “schemes”, making it more accessible to the non-expert. There is extensive coverage of plane curves, including elliptic curves and complex tori, moduli questions, and applications to cryptology.