This book series, which grew out of lectures held by the author at Bonn University, is meant to serve as an introduction to modern algebraic geometry. Two thirds of the first of the two volumes contains a detailed introduction to homological algebra, cohomology of groups, cohomology of sheaves and algebraic topology. Only the last chapter describes applications of this machinery to the algebraic geometry of compact Riemann surfaces. It basically covers classical material known from the 19th century, in particular the divisor class group (the Picard group) and the theory of Abelian varieties. The book is aimed at students with a minimal background knowledge of analysis, algebra and set theory, and the material and techniques are developed carefully from very basic notions.