Subrings of number fields, especially those that are of finite type, are worth being investigated from analgorithmic point of view. The question of how such rings are best represented in algorithms already presents several problems that challenge our theoretical understanding. In the lecture, both new results and open problems will be discussed. It is based on work that was done jointly with Daanvan Gent, Samuel Tiersma, and Jeroen Thuijs, all at Leiden.[-]