A Hindu nationalist from the Bharatiya Janata Party, Vajpayee had been hospitalised since June with a kidney infection and chest congestion. He suffered a stroke in 2009.
Prime Minister Narendra Modi was among the mourners who walked under a hot sun behind a flower-decked gun carriage carrying Vajpayee's body to a cremation ground on the banks of the Yamuna river.
In May 1998, with barely two months in office as prime minister, Vajpayee presided over the Pokhran-II nuclear tests, in which India detonated nuclear bombs in the Rajasthani desert. He sparked an arms race with Pakistan, but later launched peace talks and a cross-border bus service between Lahore and Delhi.
He served three times as India's prime minister; for 13 days in 1996, for 11 months from 1998 to 1999, and then from 1998 to 2004. The tests confirmed India as a nuclear power.
ISLAMABAD: PPP chairman Bilawal Bhutto Zardari has expressed his honest condolences on the death of former Prime Minister of India Atal Behari Vajpayee. Among the key leaders who paid their tributes were President Ramnath Kovind, Prime Minister Narendra Modi, Former President Pranab Mukherjee, former Prime Minister Dr Manmohan Singh, Home Minister Rajnath Singh, External Minister Sushma Swaraj, Murli Manohar Joshi and BJP President Amit Shah.
In another far-reaching impact of that 2000 visit, Prime Minister Vajpayee had described India and the United States as "natural allies" in a speech at a NY think-tank.
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India and the United States would go on to sign a landmark civil nuclear deal in 2008 under the watch of a different Indian prime minister, Manmohan Singh, in an initiative that President Bush had pushed relentlessly, according to several former U.S. officials. "He will be remembered for his huge contribution in bolstering US-India relations". He gave the go-ahead for underground nuclear tests in 1998 believing nuclear weapons would be deterrent against China and Pakistan.
"It was Atal Ji's exemplary leadership that set the foundations for a strong, prosperous and inclusive India in the 21st century", Modi said in a tweet.
Vajpayee's public response to the riots reflected his many contradictions. His condition is "very critical", a statement from the AIIMS says.
The government has declared a seven-day mourning period for him.
West Bengal chief minister Mamata Banerjee and Tripura chief minister Biplab Deb travelled to Delhi on Thursday evening to pay their last respects to the BJP leader.