These lecture notes grew out of a one semester introductory course on elliptic curves given to an audience of computer science and mathematics students, and assume only minimal background knowledge. After having covered basic analytic and algebraic aspects, putting special emphasis on explaining the interplay between algebraic and analytic formulas, they go on to some more specialized topics. These include the \(j\)function from an algebraic and analytic perspective, a discussion of elliptic curves over finite fields, derivation of recursion formulas for the division polynomials, the algebraic structure of the torsion points of an elliptic curve, complex multiplication, and modular forms. In an effort to motivate basic problems the book starts very slowly but considers some aspects such as modular forms of higher level which are not usually treated. It presents more than 100 exercises and a Mathematica ^{TM} notebook that treats a number of calculations involving elliptic curves. The book is aimed at students of mathematics with a general interest in elliptic curves but also at students of computer science interested in their cryptographic aspects. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Undergraduate and graduate students interested in elliptic curves. Table of Contents  Elliptic integrals
 Elliptic curves
 Elliptic functions
 A projective interlude
 The group structure on an elliptic curve
 Equivalence
 Formulaire
 Finite fields
 Division polynomials
 Torsion points
 Lattice inclusions
 Modular forms
 Hints to exercises
 Solutions to exercises
