But it is in the US that RBS's empire-building, under its ambitious chief executive, Sir Fred Goodwin, has been most evident. The bank is the eighth-biggest in the country, with 30,000 US employees and rapidly growing corporate and retail banking businesses.
In its half-year results last week, RBS said the US, where its brands include Citizens, Charter One and RBS Greenwich Capital, was one of the main drivers of its earnings, contributing 27 per cent of operating profits.
RBS has signalled widely that it wants to grow even further in the US. Sir Fred said conditions in the world's largest economy "really feel pretty good at the moment".
In a statement which is likely to pique the interest of investors, he said there was "a whole raft of things going on" with RBS in the US, and added that it was "a subject you will hear more about".
RBS has been preparing the ground for an accelerated expansion with a multimillion-dollar marketing campaign to raise the profile of its brand in the US. Its "Make It Happen" slogan has blanketed New York's John F Kennedy and La Guardia airports and adverts have swamped the television for the past few months. It sponsors major golfing tournaments and is the official backer of the US golfer Jack Nicklaus.
While most of its British rivals, including NatWest and Barclays, made mistakes that were in some cases fatal by expanding in the US, RBS's missteps have been few.
James Leal, an analyst at Teather & Greenwood, said: "RBS's experience provides the exception to the rule. The US has been a good source of growth and Citizens has in the past 15 years been an acquisitions machine."
In 1988, RBS bought Citizens, a retail banking business which was at the time the seventh-largest bank based in Rhode Island, the US's smallest state. Now Citizens has $137bn (£76bn) in assets, boosted by last year's $10bn acquisition of Charter One, a basic savings and loans institution in the Mid West.
In a country where most lenders are local and almost none have a national franchise, Citizens, under its chief executive Larry Fish, is the eighth-biggest bank in the US and has a presence in 13 out of the nation's 50 states.
The bank has paid just as much attention to building its corporate banking business in the US, though it has been doing it more quietly. In an ironic twist of fate, RBS gained a non-retail banking business in the US - Greenwich - in 2000 through its bitterly fought acquisition of NatWest, which partly fell to its smaller Scottish rival because it had been weakened by its US losses.
RBS has built on Greenwich, which focuses on selling government bonds, and asset-backed securities such as mortgages. Last year, RBS laid out significant amounts of money adding a debt capital markets business, poaching a high-profile banker, John Walsh, from Credit Suisse First Boston, to head up the business and building up a team of about 40.
RBS has added foreign exchange, syndicated loans and project financing - lending money particularly to ventures in the energy and property sectors. It has built up significant market shares in its chosen areas, this year boasting the No 1 slot in underwriting US government treasury bonds and mortgage-backed securities.
RBS wants to consolidate all of this by becoming a prominent corporate bank, boosting its business clients and offering them products ranging from basic loans to credit derivatives. Johnny Cameron, the head of its corporate banking and financial markets (CBFM), said last week there were only "three serious corporate banks" in the US. He observed: "A country the size of America needs more than three."
RBS is remaining tight-lipped at the moment about exactly how it aims to scramble up to the heights of such major corporate banking players as Bank of New York and Wachovia. But Mr Cameron did tell analysts that "all will be revealed" on 3 October, when RBS will update investors generally about its future plans.
Those plans are thought to include far more integration of Citizens and CBFM. It has added 250 business banking experts to Citizens' branches, in the hope they will be able to use this base to sell products to small and medium-sized businesses, especially in areas such as Chicago and Detroit gained through the Charter One deal.
The expensive marketing budget is also thought to be an attempt to capture larger business clients, who might not be enticed by the idea of using Citizens, still a parochial lender, but might by attracted by RBS's international reach. Indeed, such are the scale of the changes that RBS's audit committee is understood to be planning a visit to the US in the autumn to scrutinise its finances and the integration of Citizens with the CBFM arm.
If RBS is becoming increasingly bold in its ambitions, there are sceptics who wonder whether it can take the next step in the US, bringing it into the same league as banks such as Citigroup and JP Morgan Chase. RBS said it does not want to compete with them in investment banking, on the grounds that earnings from mergers and acquisitions and large equities trading operations are volatile.
It also faces stiff competition in corporate banking from some home-grown rivals. HSBC not only has a business that is about the same size as RBS's in the US, but also a much larger presence in places such as Canada and Latin America, where many potential US clients want to trade and therefore might need a bank. RBS is also competing against Barclays, which has recovered from past US débâcles to build a significant presence through its fixed-income arm, Barclays Capital, and also Barclays Global Investors, its asset management business, in the US.
Another concern is that unlike HSBC and Barclays, which use their main brand for all or most of their business in the US, RBS's strategy is seen as muddled by some. Several critics doubt whether the millions it is spending on raising the profile of the RBS brand will really make an impact in the US.
They add that RBS's decision to keep the Charter One name in the Mid West - where there is another Citizens bank - further hampers RBS's attempts to convey the impression of a seamless financial behemoth.
Ambivalence about RBS's chances might seem unfair to Sir Fred, who has shown that he can beat British banks with more long-standing international ambitions at their own game. Undoubtedly, RBS's record so far in the US is impressive, but furthering the empire will be challenging work.