This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. The two example classes of Hilbert modular surfaces and determinantal varieties are used methodically to discuss the covered techniques. For the reader the further development of the theory yields a better understanding of these fascinating objects. The text is complemented by many exercises that serve to enhance comprehension, treat additional examples, or give an outlook on further results. This book, the first of two volumes, serves as an introductory volume on schemes. The second volume concerns the cohomology of schemes. Volume I requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. A publication of Vieweg+Teubner. The AMS is exclusive distributor in North America. Vieweg+Teubner Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan. Readership Graduate students and research mathematicians interested in algebraic geometry. Table of Contents  Prevarieties
 Spectrum of a ring
 Schemes
 Fiber products
 Schemes over fields
 Local properties of schemes
 Quasicoherent modules
 Representable functors
 Separated morphisms
 Finiteness conditions
 Vector bundles
 Affine and proper morphisms
 Projective morphisms
 Flat morphisms and dimension
 Onedimensional schemes
 Examples
