The theory of dessins d'enfants (children's drawings) can be effectively used to make explicit calculations. In principle it allows one to obtain equations for ramified coverings of curves when the ramification data are specified, determine isomorphisms of certain curves or exhibit Jenkins-Strebel differentials in preferred coordinates. However, little progress has been made in utilizing the geometry/combinatorics of dessins for theoretical questions in algebraic number theory. Here we exhibit families of dessins that are geometrically simple and show that they have and/or conjecturally have interesting arithmetical properties. Speaker(s): Mehrdad Shahshahani (Sharif University of Technology, Tehran)