Update 3.30pm: Asked whether British Prime Minister Theresa May welcomed the ruling, a Downing Street spokesman said: "This is clearly an issue for the Irish Government, Apple and the European Commission."
Asked whether the British Government would like to see Apple relocate in the UK post-Brexit, the No 10 spokesman said: "The narrative from the Government has been well set out.
"Britain is open for business, we would welcome any company wishing to invest in Britain."
He stressed that all companies registered in the UK are expected to "pay the tax they owe".
Update 12.55pm: Ms Vestager has dismissed threatened court challenges from Apple and the Irish Government, saying she had a "very concrete case".
The tax rulings at the centre of the watchdog's inquiry date from 1991 and 2007.
And the probe revealed the extent of Apple's tax bills thanks to the movement of profits to subsidiaries in Ireland and a "head office" within Apple Sales International which was not based in any country, had no employees or premises and only had occasional board meetings.
It found only a small percentage of Apple Sales International's profits were taxed in Ireland and the rest was not taxed anywhere.
The commissioner highlighted 2011, when Apple Sales International recorded profits of $22bn.
Under the tax arrangement it had in Ireland, only about €50m was considered taxable, leaving €15.95bn of profits untaxed, the inquiry found.
That year, Apple Sales International paid less than €10m of corporate tax in Ireland - an effective tax rate of about 0.05% despite the headline rate being 12.5%.
The Commission said that, in subsequent years, Apple Sales International's recorded profits continued to increase but the profits considered taxable in Ireland under the terms of the tax ruling did not.
The arrangement was terminated last year when Apple Sales International and Apple Operations Europe changed their structures, the inquiry found.
— BBC Breaking News (@BBCBreaking) August 30, 2016
The companies hold the rights to use Apple's intellectual property to sell and manufacture its products outside North and South America and make yearly payments to Apple in the US for research and development.
The Commission found these expenses were deducted from the profits recorded by Apple Sales International and Apple Operations Europe in Ireland each year.
It also revealed that Apple set up its sales operations in Europe in such a way that customers were buying products from Apple Sales International in Ireland rather than from the shops that physically sold them. This way Apple recorded all sales and associated profits in Ireland.
Update 12.25pm: Sinn Féin's Finance spokesperson Pearse Doherty has called for a public inquiry to establish "who facilitated the State’s sweetheart deal with Apple".
He also said the government’s decision to appeal this ruling should be put to a vote in the Dáil and his party will be bringing forward a motion to oppose such an appeal.
Mr Doherty said: “Today’s ruling has shocked even those of us who have been watching this issue with a keen eye. Up to €13billion of tax has been lost as a result of a sweetheart deal with Apple. That is a massive amount of money and we must have a public inquiry to establish who facilitated this deal.
“Given the State is chasing people who have not paid their water charges through the courts it is unimaginable that the government will appeal this ruling.
“The government should accept the EU Commission ruling and impose the correct tax bill on Apple."
The Sinn Féin TD said any decision to appeal the ruling should be put to a Dáil vote.
He said: “The argument made by government for appealing the ruling is to protect Ireland’s reputation but unfortunately the damage has already been done and an appeal has the potential to further damage our international reputation."
Update 12pm The European Union's Representative in Ireland says the tax deal put other companies here at a serious disadvantage.
Barbara Nolan is head of the office: “Well the Commission is saying that the special treatment that Apple got in Ireland allowed it to pay less tax than other companies.
“This constitutes unfair competition, think of other companies struggling to stay afloat that did not get this special favourable treatment that Apple got.”
Update 11.25am: Apple has accused the European Commission of threatening future investment and job creation on the continent and said it is confident of overturning an order to pay €13bn in back taxes to the Irish Government over a breach of state aid rules.
Apple has said this ruling is going to have a "profound and harmful" effects in European investment and job creation.
This is not good news for Ireland as Apple is about to create 1,000 jobs in Cork.
Meanwhile, Finance Minister Michael Noonan said he profoundly disagreed with the verdict.
His office said the Ireland does not do "deals" with taxpayers: "Our tax system is founded on the strict application of the law ... without exception," Mr Noonan said.
The minister said he would seek Government support to challenge the commissioner's findings in the European courts.
"This is necessary to defend the integrity of our tax system, to provide tax certainty to business, and to challenge the encroachment of EU state aid rules into the sovereign member state competence of taxation," he said.
"It is important that we send a strong message that Ireland remains an attractive and stable location of choice for long-term substantive investment."
11.15am: Competition Commissioner Margrethe Vestager has said at a press conference that Apple was using a "so-called head office with no employees, no premises and no activity" to avoid paying tax on commercial activity taking place in Ireland.
In 2014 for evey €1m of profit Apple made, they paid €50 in tax.
Speaking today after the EU Commission decision, Chairman of the Revenue Commissioners said that Revenue has cooperated fully with the Commission’s investigation.
“We have provided all relevant information and explanations to the Commission. These demonstrate that Revenue collected the full amount of tax due from Apple in accordance with Irish tax law.
The issue of international tax planning, involving mismatches between different countries’ tax rules, is well known and is the subject of the OECD BEPS Project.”
Update: 10.40am: Europe has accused Ireland of giving illegal tax benefits to Apple worth up to €13bn.
The European watchdog has said the tech giant - which denies any wrongdoing - breached EU state aid laws by channelling some of its profits through Ireland.
A three-year investigation by Competition Commissioner Margrethe Vestager found the arrangements dating back to the early 1990s were illegal under state aid rules.
"Ireland did give favourable tax treatment to Apple."
The Revenue Commisioners have been told to recover the unpaid billions.
"Member states cannot give tax benefits to selected companies - this is illegal under EU state aid rules," the commissioner said.
"The Commission's investigation concluded that Ireland granted illegal tax benefits to Apple, which enabled it to pay substantially less tax than other businesses over many years."
BREAKING: @EU_Commission has concluded that Ireland granted undue tax benefits of up to €13 billion to Apple. News conference due in 20 mins— Tony Connelly (@tconnellyRTE) August 30, 2016
EU regulators say that deal effectively left it with a corporate tax rate of less than 1%.
The firm and the Irish government both deny doing anything wrong.
Minister Noonan is to seek Government approval to appeal the ruling and Apple will also appeal the ruling.
In a statement released this morning the Irish Government have said they "disagree profoundly" with the Commission's analysis.
Earlier: Ireland will today be ordered to recoup billions of euro in tax arrears from Apple.
The European Commission is set to declare that Ireland allowed the firm to avoid corporate tax by agreeing a special tax deal.
The ruling due this morning has been three years in the making - and while it may be what Ireland expected, it is not what the government wanted.
Competition commissioner Margarethe Vestager will publish the findings of a long-running inquiry into Apple’s tax status in Ireland.
And the Commission will claim that the company reached a private deal with the Revenue Commissioners - in short, negotiating its corporate tax rate when other firms did not have the same luxury.
Both Ireland and Apple deny the claims and both have said they will appeal.
But in the meantime Ireland will be told to reassess Apple’s tax bill - which could see the technology giant hand over billions in taxes going back years.
That money will be kept to one side while the two parties appeal to the European courts.
Daniel Shaviero, Professor of Taxation at NYU outlines what the ruling may mean for our tax laws: “Ireland does have a 12.5% tax rate in a sense what the European Comission would be saying if this continues to go through is you guys really have to charge 12.5% in other words, without having special deals.”