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It is the first time India has carried out such tests since 1974.
The experiments took place without any warning to the international community, and there has been widespread outrage and concern over the move.
The test site, in Pokhran in the northern desert state of Rajasthan, is only about 150km (93 miles) from the border with Pakistan.
The two countries have fought three wars since independence with Britain in 1947, mainly over the disputed territories of Jammu and Kashmir, and there are fears the tests could escalate the conflict and spark a regional nuclear arms race.
Prime Minister Atal Behari Vajpayee broke the news to journalists in a hastily-convened news conference.
"These were contained explosions like the experiment conducted in May 1974," he said.
"I warmly congratulate the scientists and engineers who have carried out these successful tests."
He said the devices tested were a fission device, a low-yield device and a thermonuclear device. He said there had been no release of radioactivity into the atmosphere.
There was immediate condemnation of the tests from Pakistan.
The Prime Minister of Pakistan, Nawaz Sharif, said his country had been trying to draw the world's attention to India's nuclear programme.
"As a sovereign and independent nation," he said, "Pakistan will make its own decision on the steps to be taken towards its sovereignty and defence."
Arms race threat
There was already strong diplomatic pressure on Pakistan to show restraint and avoid retaliating - the only hope of avoiding a damaging arms race.
Last month Pakistan tested its own long-range nuclear missile, the Hatf-5, which has a range of up to 1,500km (932 miles).
The tests are said to have contributed to India's decision to launch its own nuclear tests today.
India is known to have developed a short-range nuclear missile, the Prithvi, with a range of 150km (93 miles) but it is working on a much longer range system, the Agni, which could reach targets up to 2,500km (1553 miles) away.
India, Pakistan and Israel are the three nations widely suspected of nuclear capability which have not joined the 1970 nuclear non-proliferation treaty, now observed by 185 countries.
Two days later, India carried out two further nuclear tests and announced it had finished its testing programme.
Pakistan retaliated on 28 May with its own nuclear tests, confirming fears of an arms race in the region.
India's decision to carry out the tests brought a storm of international condemnation.
It also badly damaged its relationship with the United States, which imposed economic sanctions, as did several other countries.
In February 1999, the tension eased when India and Pakistan signed the Lahore accord, pledging to "resolve all issues", including that of Jammu and Kashmir.
The truce was short-lived, and hostilities over Kashmir broke out again within three months.
In October 1999, General Pervez Musharraf seized control of Pakistan in a military coup.
The change of government did not signal any improvement in relations with India, and in 2002 the two neighbours again came close to all-out war.
India and Pakistan began talking again in April 2003, and their relationship has continued to improve ever since.
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