It seems to be widely believed that the Graph Isomorphism problem in general case would not be NP-complete. The results that the problem is low for Σ^stackP_stack2 and for PP have supported this belief. Furthermore, it is also known that the problem is polynomial-time equivalent to its counting problem. This result also supported the belief since it is conversely believed that any NP-complete problem would not have this property. On the other hand, it is unknown whether the problem can be solved in deterministic polynomial-time. From these situations, it is widely believed that the problem seems to have an intermediate complexity between deterministic polynomial-time and NP-completeness. In this talk, I will first give a breif survey on a current status of the computational complexity of Graph Isomorphism problem.