The book, which is released this month, carries the warning "adult supervision recommended for minors", and is described as "scandalous satire" by its publishers.
It includes graphic illustrations of Bible characters having sexual intercourse, and other scenes depicting naked men and women as well as "gratuitous" depictions of violence.
Crumb, the book's author, is most famous for his creation Fritz the Cat, a sexually graphic "underground" comic strip. It was turned into a film that became the first animation to receive an X rating.
He has said he does not believe that the Bible is the word of God. "I take it all for myth from start to finish, with probably some faint relation to historical reality." he said.
"They're great stories. But for people to take texts as something sacred, handed down from God... that's pretty backward, I think."
The Book of Genesis illustrated by R. Crumb has been criticized by leading religious groups such as the Christian Institute.
"It is turning the Bible into titillation," said Mike Judge, of the Christian Institute, a religious think-tank. "It seems wholly inappropriate for what is essentially God's rescue plan for mankind.
"If you are going to publish your own version of the Bible it must be done with a great deal of sensitivity. The Bible is a very important text to many many people and should be treated with the respect it deserves.
"Representing it in your own way is all very well and good but it must be remembered that it is a matter of people's faith, their religion.
"Faith is such an important part of people's lives that one must remember to tread very carefully."
Other leading religious figures have been more supportive of the work. "I didn't think it was satire," said the Bishop of Croydon, the Rt Rev Nick Baines.
"He set out to say; 'this is important, fundamental myth' and it seems to me he's done a good job."