AbstractThe distinguishing number of a graph G is the smallest positive integer r suchthat G has a labeling of its vertices with r labels for which there is no non-trivialautomorphism of G preserving these labels.In early work, Michael Albertson and Karen Collins computed the distinguishingnumber for various finite graphs, and more recently Wilfried Imrich, Sandi Klavˇzarand Vladimir Trofimov computed the distinguishing number of some infinite graphs,showing in particular that the Random Graph has distinguishing number 2.We compute the distinguishing number of various other finite and countablehomogeneous structures, including undirected and directed graphs, and posets. Weshow that this number is in most cases two or infinite, and besides a few exceptionsconjecture that this is so for all primitive homogeneous countable structures...