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The National Gallery was the second-most visited attraction in the UK, with 6.03 million people coming through its doors (up 14% on 2012).
The Natural History Museum came in third place (up 6.7% to 5.35 million).
And several other London attractions also saw big increases, including Kew Gardens (up 29%) London Zoo (up 26.4%) the Tower of London (up 18.4%) and the National Maritime Museum at Greenwich (up 27%).
Alva director Bernard Donoghue said: "2014 is sure to be a momentous year, seeing the 450th anniversary of the birth of Shakespeare.
"Therefore we are delighted that Shakespeare's Globe in London and the Royal Shakespeare Company in Stratford-upon-Avon, along with the South Bank Centre in London, are Alva's newest members."
A large increase in visitors at tourist attractions across the capital is being put down to last year's fine summer.
Figures from the Association of Leading Visitor Attractions (ALVA) show a 12% increase in 2013 - double the rise seen across the rest of the UK.
The British Museum was named the most popular visitor attraction in the country for the seventh year running. It had its most successful year on record, with a 20% increase in its visitor numbers to £6.7 million.
A spokesman for ALVA said: "Exhibitions such as 'Life and death in Pompeii and Herculaneum' undoubtedly had an impact on numbers, as did the success of 'Ice Age Art: arrival of the modern mind' and 'Shunga: sex and pleasure in Japanese Art'.",
The British Museum has been named London's most popular attraction according to new figures from the Association of Leading Visitor Attractions. 6.7 million people visited its exhibitions last year, a rise of 20%.
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Here are the UK's top 10 tourist attractions, and the number of people who visit them. They're all in London.