  The distinguishing number of a graph $G$ is the smallest positive integer $r$
such that $G$ has a labeling of its vertices with $r$ labels for which there is
no non-trivial automorphism of $G$ preserving these labels. Albertson and
Collins computed the distinguishing number for various finite graphs, and
Imrich, Klav\v{z}ar and 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
countable homogeneous structures, including undirected and directed graphs, and
posets. We show that this number is in most cases two or infinite, and besides
a few exceptions conjecture that this is so for all primitive homogeneous
countable structures.
