We shall explain how certain kinds of subgroups of Coxeter groups can be used to derive regular and semi-regular tessellations. In particular, we shall briefly describe the construction of chiral polytopes. Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of hyperbolic n-space. For small n, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of the hyperbolic n-space. We shall show how linear fractional transformations over rings of rational and quadratic integers are related to the symmetry groups of regular tessellations of hyperbolic plane or 3-space.