These lecture notes are based on a series of three lectures given at the NATO Advanced Study Institute on New Challenges in Digital Communications in Vlora, Albania, 2008. The goal is to provide an elementary approach to elliptic and hyperelliptic curve cryptography with emphasis on the underlying mathematical problems. The first lecture entitled “Computational number theory, cryptography, and curves” provides selected problems from basic cryptographic schemes and shows why cryptography and computational number theory are so intertwined. In the second lecture entitled “Some (new) attacks to the elliptic curve discrete logarithm problem”, various attacks are presented and discussed. The last lecture is entitled “Real hyperelliptic curves”. It describes the use of hyperelliptic curves in cryptography and is a more specialized topic covering various new results on the real model of hyperelliptic curves. We only provide the lecture notes for the first lecture in this contribution. The other lecture notes will be made available through other journal publications.