A duke is the senior rank in the English peerage. The word is derived from the Latin dux, meaning leader: thus William the Bastard was titled as Ducis Normannorum et regis Anglorum -- Duke or senior man of Normandy and King of England. The area ruled by a duke or duchess is a Duchy or Dukedom.
The premier English dukedom is Cornwall, but that is an appendage of the Crown; the first non-royal dukedom was granted in 1351 to Henry, Earl of Lancaster, Derby, Lincoln and Leicester; the premier surviving non-royal dukedom is Norfolk, first created in 1397.
Dukes in the SCA
In the SCA this title is awarded to someone who has completed a reign over a kingdom two or more times. The ducal coronet displays strawberry leaves and dukes are permitted to have a ducal coronet on their arms. The honorific for a duke is "Your Grace".
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The Venetian title of doge is also derived from the latin dux, and would therefore nominally equivalate to Duke.