The wife of a duke is known as a duchess, which is also the title of a woman who holds a dukedom in her own right, referred to as a duchess suo jure; her husband, however, does not receive any title. In the order of precedence in the United Kingdom, non-royal dukes without state offices or positions generally take precedence before all other nobility, in order of date of creation, but after royalty and certain officers of state.
With the exceptions of the dukedoms of Cornwall and Rothesay (which can be held by only the eldest son of the Sovereign who is also heir apparent) as well as last creation of the dukedom of Edinburgh (which is a life peerage that will become extinct on the death of the current Duke), royal dukedoms are hereditary, according to the terms of the letters patent that created them, which usually contain the standard remainder to the "heirs male of his body lawfully begotten." The British monarch also holds and is entitled to the revenues of the Duchy of Lancaster, and within the borders of the County Palatine of Lancashire is by tradition saluted as "The Duke of Lancaster" even though the title is technically extinct, while, in the Channel Islands, the monarch is The Duke of Normandy. Even when the monarch is a Queen regnant, she does not use the title of Duchess.
Non-royal dukedoms
There are the following extant non-royal dukes in the United Kingdom (in alphabetical order):
Speak to as: Your Grace (formal and employees), Duke (social)
Ceremonial, formal, or legal title: The Most High, Noble and Potent Prince His Grace [forename], Duke of _____.
Coronet
A British or Irish duke is entitled to a coronet (a silver-gilt circlet, chased as jewelled but not actually gemmed) bearing eight conventional strawberry leaves on the rim of the circlet. The physical coronet is worn only at coronations. Any peer can bear his coronet of rank on his coat of arms above the shield.