One of the most fundamental questions one can ask about a vertex-transitive graph 􀀀 is what is the full automorphism group Aut(􀀀) of 􀀀? This is usually a difficult question, as is evidenced by determining Aut(􀀀) should allow for the solution of many other problems concerning 􀀀. For example, is 􀀀 edge-transitive, 1/2-transitive, or normal? Additionally, one should be able to solve the isomorphism problem for 􀀀. In this talk, we will report on recent progress in determining Aut(􀀀), for 􀀀 in certain classes of vertex-transitive graphs, especially Cayley graph of certain abelian groups.