This is the first of three volumes on algebraic geometry. Today, most algebraic geometers are well-versed in the language of schemes, but many newcomers are still initially hesitant about them. Ueno's book provides an inviting introduction to the theory, which should overcome any such impediment to learning this rich subject.The book begins with a description of the standard theory of algebraic varieties. Next, sheaves are introduced and studied, using as few prerequisites as possible, and finally, Ueno describes schemes and their properties. Further properties of schemes will be discussed in the second volume. Ueno's book is a self-contained introduction to this important circle of ideas which assumes only a knowledge of basic notions from abstract algebra (such as prime ideals).