We want to give an overview on arithmetical aspects of abelian varieties and their torsion structures, isogenies, and resulting Galois representations. This is a wide and deep territory with a huge amount of research activity and exciting results ranging from the highlights of pure mathematics like the proof of Fermat’s last theorem to stunning applications to public-key cryptography. Necessarily we have to be rather superficial, and thus specialists in the different aspects of the topics may be disappointed. But I hope that for many, and in particular for young researchers, the chapter may serve as an appetizer and will raise interest for a fascinating area of mathematics with many open problems (some are very hard and worth a Fields Medal but others are rather accessible).