A Cayley graph Cay(G, S) of a group G is called a CI-graph if whenever T is another subset of G for which Cay(G, S) congruent to Cay(G, T), there exists an automorphism sigma of G such that S-sigma = T. For a positive integer m, the group G is said to have the m-CI property if all Cayley graphs of G of valency m are CI-graphs; further, if G has the R-CI property for all k less than or equal to m, then G is called an m-CI-group, and a \G\-CI-group G is called a CI-group. In this paper, we prove that A(5) is not a 5-CI-group, that SL(2, 5) is not a 6-CI-group, and that all finite 6-CI-groups are soluble. Then we show that a nonabelian simple group has the 4-CI property if and only if it is Ag, and that no nonabelian simple group has the 5-CI property. Also we give nine new examples of CI-groups of small order, which were found to be CI-groups with the assistance of a computer. (C) 1998 Academic Press.