Mr Kalam served as India's 11th president from 2002 to 2007 and was popularly known as "Missile Man" after pioneering the country's military missile programme.
He died on Monday after collapsing as he delivered a scientific lecture.
India has declared seven days of national mourning. The former professor's body will be flown to New Delhi on Tuesday for burial.
"His death is a great loss to the scientific community," Prime Minister Narendra Modi said. "He took India to great heights. He showed the way."
Indian President Pranab Mukherjee said: "Kalam would be long remembered for his passion for science and innovation and his contribution as an eminent scientist, administrator, educationist and writer."
APJ Abdul Kalam was born in 1931 into a middle-class family in Rameshwaram, a town well-known for its Hindu shrines in the southern state of Tamil Nadu.
His father owned boats which he rented out to local fishermen, but he himself began his career as a newspaper vendor.
He then earned a degree in aeronautical engineering from a technology institute in Chennai (Madras).
Mr Kalam joined the Vikram Sarabhai Space Centre in the neighbouring state of Kerala in the 1960s as one of its first three engineers.
He played a major role in the centre's evolution to a key hub of space research in India, helping to develop the country's first indigenous satellite-launch vehicle.
He worked for the Defence Research and Development Organization and the Indian Space Research Organization.
Indian scientists have hailed him as the father of the Indian nuclear bomb and its missile delivery systems.
He also played a key role when India tested its nuclear weapons in 1998.
The former president was proud of his Indian education and liked to describe himself as "Made in India", having never been trained abroad.
Former Indian President APJ Abdul Kalam has died at the age of 83.