Grothendieck's theory of schemes has proved to be a spectacularly successful foundation for algebraic geometry, providing the framework for the understanding and solution of many classical problems. It has also established a unification between geometry and number theory--"arithmetic geometry''--that has been equally spectacular, leading to solutions of the Weil conjectures, the Mordell conjecture, and the proof of Fermat's Last Theorem. My goal in this course will be to provide an introduction to scheme theory, preparing both for work both in "classical'' algebraic geometry over the complex numbers and in the "arithmetic'' theory of schemes over rings of integers in numberfields. Hartshorne's classic text will be our main guide, but I plan to supplement it with other more arithmetic treatments, including Grothendieck's EGA. I will skip chapter I of Hartshorne's book, beginning instead by discussing affine algebraic sets and spaces; then I will go straight to chapter II. Students should have a good foundation in commutative algebra, as well as some experience with global techniques in geometry (e.g. differential or algebraic topology).