In this chapter we study holomorphic line bundles on complex tori, i.e., quotients of complex vector spaces by integral lattices. The main result is an explicit description of the group of isomorphism classes of holomorphic line bundles on a complex torus T. The topological type of a complex line bundle L on T is determined by its first Chern class c1 (L) ∈ H2(T, ℤ). This cohomology class can be interpreted as a skew-symmetric bilinear form E : Γ × Γ → ℤ where Γ = H1(T, ℤ), is the lattice corresponding to T. The existence of a holomorphic structure on L is equivalent to the compatibility of E with the complex structure on Γ ⊗ ℝ by which we mean the identity E (iv, iv′) = E(v, v′). On the other hand, the group of isomorphism classes of topologically trivial holomorphic line bundles on T can be easily identified with the dual torus Tv = Hom(Γ, U(1)). Now the set of isomorphism classes of holomorphic line bundles on T with the fixed first Chern class E is a Tv-torsor. It can be identified with the Tv-torsor of quadratic maps α : Γ → U(1) whose associated bilinear map Γ × Γ U(1) is equal to exp(π i E). These results provide a crucial link between the theory of theta functions and geometry that will play an important role throughout the first part of this book.