You might want to look at section 3 of D1.1; Table 3.1 lists the ASTM designations of steels commonly used for structural applications.
AISC's Code of Standard Practice (which I think is still available on the aisc.org website as a free download) explains who has responsibility for what. If you are involved with structural steel as a the owner, the steel supplier, the fabricator, the erector, the painter, or the inspector, you really ought to read through that.
As was advised by John and hogan, the contract should specify which materials are to be used.