This new textbook is based upon the notes that accompany the author’s graduate course “Algebraic Geometry I” at Tsinghua University.Much of commutative algebra owes its existence to algebraic geometry and vice versa, and this is why there is no clear border between the two. In learning algebraic geometry, you not only learn more commutative algebra, but also develop a geometrical way of thinking about it.Eduard J.N. Looijenga is one of the best algebraic geometers in the world and a leading mathematician of his generation. He is a member of the Royal Netherlands Academy of Arts and Sciences and currently teaches at Tsinghua University.