We will define a notion of topological similarity between tuples of elements in a Polish group. This relation is coarser than conjugacy relation, and is usually easier to deal with. We then discuss typical examples of groups with large and small classes of similarity. One of the central objects will be the group of isometries of the Urysohn space. We will mention some new results on extensions of partial isometries in this space, and illustrate their connection with conjugacy classes and classes of topological similarity.