Some recent results on graphs with a high degree of symmetry

A graph G is said to be s-set-homogeneous (s-CH) (respectively s-connected-set-homogeneous (s-CSH)) if for any two isomorphic (respectively connected) induced subgraphs of G of order at most s there exists an automorphism of G that sends one to the other. We say that the graph G is s-homogeneous (respectively s-connected homogeneous (s-CH)) if any isomorphism between (respectively connected) induced subgraphs of order at most s extends to an automorphism of the whole graph. In this talk, I will survey some old and new results in this area.