Is the area called Blackmoor in the Greyhawk setting any relation to the old Blackmoor campaign setting, or are they unrelated? I know very little about either...
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They're related, but they're not the same. Gary Gygax and Rob Kuntz once played a game in Dave Arneson's Blackmoor campaign (with their player characters, Mordenkainen and Robilar). Later he included references to Blackmoor and the City of the Gods in the World of Greyhawk boxed set, because his (mostly) Greyhawk PCs had been there.
Greyhawk's Blackmoor doesn't really look much like Dave Arneson's Blackmoor, although they share a few names.
For more about Blackmoor, see Havard's website, as he's the expert.