There are approximately only 120 lines of dialogue in the whole film.
One-Eye was a nickname for Oden (Odin), who in the Norse legends gave one of his eyes as a sacrifice to the well of Mimer to get wisdom. The sacrifice also gave him the power to see hidden things.
During the entire "Men of God" sequence, the sea is made of black plastic bags.
Mads Mikkelsen took on the lead role, thinking that it would be quite an easy job for him, seeing as it had no dialogue. He didn't realize the arduous physical demands of shooting in forbidding Scottish locations.
The idea of One-Eye being a mute came to Nicolas Winding Refn in a dream where One-Eye could not talk. Once he woke up, he phoned Roy Jacobsen and told him about the idea.