This book is a translation from the French of an expanded version of a graduate course given in 1997. The author states as its goal to offer an elementary introduction to the classical theory of complex tori while presenting in parallel a more "modern" point of view using more recent theories (sheaves, Chern classes, cohomology, etc.). Now this goal is attained: The text is skillfully streamlined so that, to cover a large amount of important and different topics, it needs only a one-page bibliography. To keep the discussion smooth, some more complicated definitions and proofs are (mainly in the last chapters) referred to the literature and/or explained in footnotes, which also contain more references than the bibliography at the end of the book. Every chapter has at its end several interesting exercises. The author treats in the first two chapters the classical theory of one-dimensional complex tori, elliptic curves and their moduli, and then states as his purpose to generalize these results to arbitrary dimension. Thus, in the next chapters the classical theory of complex tori is presented, intertwined with the now usual tools from analytic and algebraic geometry, i.e., differential forms and de Rham cohomology (in Chapter 3), theta functions and divisors (Chapter 4), line bundles, sheaf cohomology, and first Chern class (Chapter 5). Among others, we find as main results Weil's theorem that any effective divisor on a complex torus is the divisor of a theta function (Theorem 4.9), and the Appell-Humbert theorem parametrizing the line bundles on a complex torus by their types (Theorem 5.17). In Chapter 6 abelian varieties are introduced (in relation to Riemann conditions) and their projective imbedding is presented together with (among others) the standard facts about their field of meromorphic functions and their endomorphisms. Chapter 7 treats moduli spaces of polarized abelian varieties using Riemann's theta functions. We see projective embeddings of the moduli spaces via theta constants. Finally, Chapter 8 assembles several newer non-standard results concerning subvarieties of a complex torus. We find connectedness results and discussions of fundamental groups. Here the text is no longer as elementary as in the beginning and relies on more recent papers cited in footnotes. As examples, we indicate the Conjecture in 8.3: Let $X$ be a complex torus, and let $A$ be an irreducible nondegenerate subvariety of $X$ which is a local complete intersection. We have $$\pi_j(A)\simeq\pi_j(X)\quad \text{for} j\leq 2\dim A-\dim X.$$ Another example is the final Corollary 15 saying that the canonical bundle of a smooth subvariety of a complex torus that is invariant under translation by any nontrivial torus is ample.