In our talk, a highly symmetric graph will mean a vertex-transitive graph whose automorphism group is much larger than the order of the graph. We will introduce two numerical parameters – the Cayley and quasi-Cayley deficiency – that will allow us to divide the class of highly symmetric graphs into disjoint subfamilies. We will discuss several known families of highly symmetric graphs, determine their Cayley and quasi-Cayley deficiencies, and try to ask the question which graphs are extremal with regard to these parameters.