Homogenous structures exhibit a high degree of symmetry. In particular their automorphism group is transitive, and any partial isomorphism between two finite substructures extends to an automorphism of the entire structure. It is thus natural to better understand these symmetries, and one approach is by trying to break them. The distinguishing number provides an interesting tool to do so and at the same time providing structural information. Two such well known homogeneous structures are the rationals and the Rado graph. The first one is easily seen to have unbreakable symmetry in this setting, its distinguishing number is infinite. On the other hand Imrich et al. showed that the Rado graph has distinguishing number 2. We will present an overview of the distinguishing number of the homogeneous simple and directed graphs through their classification, and discuss recent results for the countable Urysohn homogenous metric spaces of given spectrum.