The Royal Bank of Scotland, like the Bank of England, began when a group of holders of public debt received a royal charter to incorporate as a banking institution. Parliament granted the royal charter creating Scotland’s second public bank on May 31, 1727. The Scottish Parliament had chartered Scotland’s first
public bank, the Bank of Scotland, on July 17, 1695, before the unification of Scotland and England. In the Rebellion of 1715, the Bank of Scotland had appeared to stand on the Stuart side, a point frequently recalled by those who wanted to create a rival bank, the Royal Bank of Scotland. Parliament later chartered a third Scottish public bank, and also encouraged the growth of private banks. Scotland promoted competition among banks to a much greater extent than England, pioneering the development of free banking, which flourished in the United States in the first half of the 19th century.
The Royal Bank of Scotland earned a place in the history of money and banking when it developed the antecedents of overdraft privileges. In his famous book, An Inquiry into the Nature and Causes of the Wealth of Nations, perhaps the most famous book on economics, Adam Smith attributed this important banking innovation to the public banks of Scotland. Although he does not credit a single bank for the innovation, his description captures the spirit of the innovation in the language of the day:

They invented, therefore, another method of issuing their promissory notes; by granting what they called cash accounts, that is by giving credit to the extent of a certain sum (two or three thousand pounds, for example) to any individual who could procure two persons of undoubted credit and good landed estate to become surety for him, that whatever money should be advanced to him, within the sum for which the credit had been given, should be paid on demand, together with the legal interest. Credits of this kind are, I believe, commonly granted by banks and bankers in all different parts of the world. But the easy terms upon which the Scottish banking companies accept of repayment, are so far as I know, peculiar to them, and have, perhaps, been the principal
cause, both of the great trade of those companies and of the benefit which the country has received from it. (Smith, 1937, pp. 282)
Other authors, such as Glyn Davies, confer the credit for this innovation to the Royal Bank of Scotland, and cite this innovation as the beginning of the flexible overdraft. The Royal Bank of Scotland (now called the Royal Bank of Scotland, Limited) remains one of the leading commercial banks of Scotland. In 2008, it
became one of the leading beneficiaries of the United Kingdom’s plan to bailout banks who had overinvested in toxic assets. In 2009, it became officially classified as a public-sector entity, because the United Kingdom’s
government had absorbed such a large share of its liabilities (MacDonald and Norman, February 2009).
See also: Bank of Scotland, Scottish Banking Act of 1765 

Checkland, S. G. 1975. Scottish Banking: A History, 1695–1973.
Davies, Glyn. 1994. The History of Money. 
MacDonald, Alistair, and Laurence Norman. “World News: Bank Bailouts, Sinking Revenue Fray U.K.’s Ledger.” Wall Street Journal (Eastern Edition), February 20, 2009, p. A10.
Smith, Adam. 1937. An Inquiry into the Nature and Causes of the Wealth of Nations.