Abelian varieties are a natural generalization of elliptic curves to higher dimensions, and additionally play a significant role in the geometric approach to modern algebraic number theory. The use of theta functions, particularly since Mumford’s work, has been an important tool in the study of abelian varieties and invertible sheaves. In this book, the author has focused on the analytical aspects of the geometry of abelian varieties, rather than taking the alternative algebraic or arithmetic points of view. His purpose is to provide an introduction to complex analytic geometry. Thus, he uses Hermitian geometry as much as possible. One distinguishing feature of the author’s presentation is the systematic use of Mumford’s theta group. This allows him to give precise results about the projective ideal of an abelian variety. In its detailed discussion of the cohomology of invertible sheaves, the book incorporates material previously found only in research articles.