The proof of Fermat's last theorem due to A. Wiles relies on the proof of the Taniyama-Shimura-Weil conjecture. The latter says that every elliptic curve defined over the rationals is modular. In this lecture I want to give a (mostly) elementary introduction into the modularity problem for algebraic varieties. In particular, I want to discuss the modularity of rigid and non-rigid Calabi-Yau varieties.