The prefix The Most Noble is a title of quality attached to the names of dukes and duchesses in the United Kingdom. It is usually used on legal documents, where a duke would be described as, for instance, The Most Noble John William, Duke of London, while the prefix His Grace (or Her Grace) is used in less formal situations.
The Most Noble is a contraction of the most formal style of a duke, The Most High, Noble and Potent Prince, which is used only in extremely formal circumstances.