Politics and Cyber Wars
Estonia says many state websites have been affected
Many of the attacks have come from Russia and are being hosted by Russian state computer servers, Tallinn says.
It wants to put the issue at the top of Friday's EU-Russia summit agenda. Moscow denies any involvement.
Estonia says the attacks began after it moved a Soviet war memorial in Tallinn. The move was condemned by the Kremlin.
The memorial's removal also triggered riots amongst mostly ethnic Russian living in Estonia, during which one person was killed and more than 150 injured.
Estonians say the memorial symbolised Soviet occupation of the Baltic state. Russians say it is a tribute to those who fought the Nazis.
The Estonian government says its state and commercial websites - including a number of banks - are being bombarded by mass requests for information - overwhelming their computer servers.
Targets of the so-called denial-of-service attacks have also included the Estonian foreign and defence ministries and leading newspapers and banks.
In some cases, officials have simply blocked access to the servers from outside Estonia, to prevent them from being attacked.
"A couple weeks ago when the whole thing started we had some problems in our online services and then our mail server was absolutely inundated with spam e-mails as well," Estonian journalist Aet Suvari told the BBC.
"In the past few weeks it has been quite difficult for some government officials to read their e-mails on the web, to get access to the banks."
The defence ministry says that the cyber attacks come from all over the world, but some have been hosted by Russian state servers.
It says that instructions on how to carry out cyber warfare are circulating in Russian on Russian websites.
Estonian Prime Minister Andrus Ansip has directly accused Russia of being responsible, pointing the blame at the Russian government.
Nato and EU internet experts are all helping to track down the culprits, but Estonian officials say that they have had no co-operation from Russia.
While technical experts say that the initial wave of attacks came from official structures in Russia, they now say it might be very difficult to track the perpetrators down, the BBC's Steven Eke says.
Russia - which has a large community of hackers and computer virus-writers - has been accused of mounting such attacks before in the US and Ukraine.
Moscow denies any involvement in the internet attacks on Estonia.
Kremlin spokesman Dmitry Peskov told the BBC that the allegations were "completely untrue".
Estonia has one of the most developed internet networks in Europe and is a pioneer in the development of government websites, which correspondents say makes it highly vulnerable to cyber attacks.
The government says the attacks could cut off its state and commercial websites from the rest of the world.
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