In the British peerage, a royal duke is a member of the British royal family, entitled to the titular dignity of prince and the style of His Royal Highness, who holds a dukedom. Dukedoms are the highest titles in the British roll of peerage, and the holders of these particular dukedoms are princes of the blood royal. The holders of the dukedoms are royal, not the titles themselves. They are titles created and bestowed on legitimate sons and male-line grandsons of the British monarch, usually upon reaching their majority or marriage. The titles can be inherited but cease to be called "royal" once they pass beyond the grandsons of a monarch. As with any peerage, once the title becomes extinct, it may subsequently be recreated by the reigning monarch at any time.
Royal status of dukedoms
In the United Kingdom, there is nothing intrinsic to any dukedom that makes it "royal". Rather, these peerages are called royal dukedoms because they are created for, and held by, members of the royal family who are entitled to the titular dignity of prince and the style Royal Highness. Although the term "royal duke" therefore has no official meaning per se, the category "Duke of the Blood Royal" was acknowledged as a rank conferring special precedence at court in the unrevoked 20th clause of the Lord Chamberlain's order of 1520. This decree accorded precedence to any peer related by blood to the sovereign above all others of the same degree within the peerage. The order did not apply within Parliament, nor did it grant precedence above the archbishop of Canterbury or other Great Officers of State such as is now enjoyed by royal dukes. But it placed junior "Dukes of the Blood Royal" above the most senior non-royal duke, junior "Earls of the Blood Royal" above the most senior non-royal earl (cf. Earldom of Wessex), etc. It did not matter how distantly related to the monarch the peers might be (presumably they ranked among each other in order of succession to the Crown). Although the 1520 order is theoretically still in effect, in fact the "Blood Royal" clause seems to have fallen into desuetude by 1917 when King George V limited the style of Royal Highness to children and male-line grandchildren of the sovereign. Thus peers of the blood royal who are neither sons nor grandsons of a sovereign are no longer accorded precedence above other peers.
Assuming that Alexander Windsor, Earl of Ulster and George Windsor, Earl of St Andrews succeed their fathers to become the third Duke of Gloucester and the third Duke of Kent respectively, their peerages (as created in 1928 and 1934) will cease to be royal dukedoms; instead their holders will become "ordinary" dukes. The third dukes of Gloucester and Kent will each be styled His Grace because, as great-grandsons of King George V, they are not princes and are not styled HRH. Similarly, upon the death of Prince Arthur, Duke of Connaught and Strathearn (1850–1942) (the third son of Queen Victoria), his only male-line grandson, Alastair, Earl of Macduff (1914–43), briefly succeeded to his peerages and was styled His Grace. Before the 1917 changes, his style had been His Highness Prince Alastair of Connaught.
Current royal dukedoms
The current royal dukedoms, held as principal titles, in order of precedence, are:
The following dukedoms are currently held by William, Prince of Wales: Duke of Cornwall is a title automatically held by the Sovereign's eldest son in England. In addition to the dukedom of Cornwall, a peerage, the holder also enjoys a life interest in the Duchy of Cornwall. Duke of Rothesay is a title automatically held by the Sovereign's heir apparent in Scotland, who is properly called "HRH The Prince William, Duke of Rothesay" (rather than "HRH The Prince of Wales") in Scotland. Duke of Cambridge is a title which was conferred on the Prince on his wedding day, 29 April 2011.
With the exceptions of the dukedoms of Cornwall and Rothesay (which can only be held by the eldest living son of the sovereign who is also the heir apparent), these dukedoms are hereditary according to the letters patent that created them. Those patents contain the standard remainder to "heirs male of his body", with the exception of the Dukedom of Edinburgh, which is a life peerage and will become extinct on the death of the current Duke.