Kalam, who occupied Rashtrapati Bhavan in 2002-07 as the 11th president, breathed his last at 7:45pm on Monday at the Bethany Hospital.
He was rushed there after collapsing at the IIM-Shillong where he was delivering a lecture on "liveable planet", officials at the premier institute told IANS.
The 84-year-old Kalam, who had been lecturing at the IIM-Shillong since last year, had reached the institute at 5:40pm.
According to its director Amitabha De, he was taken to the guesthouse where he rested for a while and came to the lecture hall at 6:40pm.
"Kalam must have barely spoken for five minutes when he suddenly collapsed," De said, recalling the tragedy. "We rushed him to the hospital by 7pm where he passed away."
Meghalaya Governor V Shanmuganathan said the doctors "made enormous efforts to save him but we lost a great leader".
Meghalaya Chief Secretary PBO Warjri broke the news to IANS: "Kalam is no more."
Doctors at the hospital said Kalam was brought "almost dead" after suffering a cardiac arrest.
Hospital Director John Sailo Ryntathiang said they did their best to revive him. But he could not be saved.
As the news spread in the city, thousands of tear-eyed people gathered at the Bethany Hospital, shouting "Kalam Amar Rahe" and followed the army ambulance that took his body to the Military hospital where it will be kept overnight.
Meghalaya Deputy Chief Minister Roytre Christopher Laloo, who was at the hospital with his colleagues, said Kalam's body will flown to New Delhi via Guwahati in an air force helicopter on Tuesday morning.
President Pranab Mukherjee, who is in Karnataka, led the nation in mourning Kalam.
Saying he will "cherish" their long association, Mukherjee said: "Dr. Kalam will be long remembered for his passion for science and innovation...."
The government has announced a seven day state mourning, until August 2, during which the national flag will flay at half-mast and there will be no official entertainment.
The date, time and venue of the state funeral will be intimated later, the union home ministry said.
Prime Minister Narendra Modi, terming himself greatly upset, said "in this great shocking situation, I have no words to say".
Vice President Hamid Ansari termed Kalam a "true son of India" and Lok Sabha Speaker Sumitra Mahajan saluted him as a "true patriot, world-renowned scientist and original thinker".
Former prime minister Manmohan Singh recalled he had worked very closely with Kalam and "greatly benefited from his advice".
Congress president Sonia Gandhi called Kalam "a scholar-statesman and one of the greatest scientific minds".
LK Advani, who was deputy prime minister when Kalam became president, said he "served Mother India literally until the last breath".
Other leaders and scientists also mourned his death, while gloom spread in his hometown Rameshwaram, a Hindu pilgrim town in coastal Tamil Nadu, 600 km from Chennai.
Kalam's early years were, however, steeped in poverty when, as a mere eight-year-old, he hawked newspapers to supplement the income of a large family.
There were times when food was scarce in the family and his hard-pressed mother stretched every resource to the utmost to keep her five sons and daughters as well as her boat owner husband and his brother's families fed, clothed and in good health.
By his own admission, Kalam would wake up much before dawn to distribute newspapers in the town after collecting newspaper bundles at the Rameshwaram railway station. The tough routine lasted a year.
His sister pawned jewellery with a moneylender so that the studious Kalam could have Rs. 600 to join the Madras Institute of Technology.
From there, Kalam went on to become one of the most celebrated aerospace and defence scientists in the country.
He was awarded with the 'Padma Bhushan' in 1981, 'Padma Vibhushan' in 1990 and ‘Bharat Ratna’, the highest civilian award of India, in 1997.
Before he became the president, he was an aerospace engineer with Defence Research and Development Organisation and Indian Space Research Organisation.
He became the head of the Defence Research and Development Organisation in 1992.
He studied aeronautical engineering and played a key role in the development of India’s first satellite launch vehicle, the SLV-3, which launched the Rohini satellite in 1980 to take India into the space club.
APJ Abdul Kalam also played a key role in building India’s strategic missile systems and in the nuclear test at Pokhran in Rajasthan in May 1998, when Atal Bihari Vajpayee was the prime minister, though of late he became an advocate of nuclear-free world.
He became the principal scientific advisor to the Indian government in 1999 with the rank of a cabinet minister. He held the post until 2001.
Kalam was known for his motivational speeches and interaction with the youths in India as part of his advocacy plan to develop India into a developed nation by 2020 that he mentioned in his book ‘India 2020’.
Until the very end, his focus had stayed on that goal and to this end he continued to follow rigorous schedule travelling across India and teaching students at IITs and IIMs, to address conferences and to meet students and people from all walks of life.
In the last two decades, he had interacted with over 18 million youths.
Kalam, who was also known as ‘People's President’, had authored 29 books, of them ‘Wings of Fire’ where he covered his life up to 1992 sold more than a million copies.
Inspired by that success, he had written another book, ‘Turning Points’, where he shared seven turning points or challenges of his life with the world.