In the 19th century Dirichlet proved a structural description of the unit group of the ring of integers in an algebraic number field. In the non-commutative case, the analogous question is unsolved. In our project we will develop and implement algorithms to investigate unit groups of orders in division algebras. We will compute explicit matrices generating this group and defining relations. This description of the unit group will be used in experiments to obtain insights into the structure of these groups and to test relevant conjectures.