Armstrong was mission commander of Apollo 11 and on July 20, 1969, along with Edwin "Buzz" Aldrin, he spent two hours on the moon's surface. His death follows heart-bypass surgery earlier this month.
Armstrong entered the history books alongside other icons of human exploration such as Christopher Columbus, and half a billion people tuned in to watch the historic moment.
He walked backwards down the ladder of the moon landing space ship Eagle, layed his left foot on the moon's surface and uttered the iconic words: "That's one small step for a man, one giant leap for mankind."
Armstrong also uttered the famous touch-down words as the lunar module hit the moon's surface: "Houston, Tranquility base, here. The Eagle has landed." Mission control responded: "Roger ... we copy you ... you got a bunch of guys about to turn blue ... We are breathing again. Thanks a lot."
The Apollo 11 mission turned out to be Armstrong's last space flight. The following year he was appointed to a desk job.
Last November, Armstrong received the Congressional Gold Medal, the highest US civilian award.
He was an extemely private man and made very few public appearances, spending much of his career as a professor of aeronautical engineering.
He was born in the town of Wapakoneta, Ohio, on August 5, 1930, but had Scottish roots through his father's side of the family.
Dumfriesshire town Langholm claimed the astronaut as one of its own and made him a Freeman in 1972. In a rare public outing that year, he paid a visit to the town. He made another pilgrimage to Scotland in 2010 when he played golf at Leven Links in Fife.
At the time, Peter Paterson, the starter at the course, said: "Neil said he liked the course and was loving his trip to Scotland. He was a perfect gentleman, very humble. It was a total honour to meet him."
Armstrong's father worked for the government and, as a child, Neil and his family were constantly on the move as he took up new positions.
Armstrong took his first flight aged six with his father and formed a passion for aeronautics.
His hero was transatlantic aviator Charles Lindbergh, who he later became close to. By the age of 16 he could fly.
Armstrong became a test pilot for the National Advisory Committee on Aeronautics, Nasa's forerunner, and served as a pilot in Korea.
With Nasa, he served as one of an elite group selected to pit technology against nature's limitations.
In 1962, John F Kennedy promised to put a man on the moon by the end of the decade. The 1969 flight was hailed as a miracle for the United States space programme, and inspired a generation. But after the voyage, Armstrong stayed on at Nasa for just two years. He left to teach engineering at the University of Cincinnati in his native Ohio.
Armstrong never gave interviews and did not write any memoirs. In 2005, he finally consented to an exhaustive biography written by the historian James Hansen.
Armstrong lived north of Cincinnati, with his second wife Carol.
A statement from Armstrong's family said he died following complications resulting from cardiovascular procedures.
Tributes flooded in for the former astronaut.
Astronomer Sir Patrick Moore said: "As the first man on the moon, he broke all records.
"I knew him well. He was a man who had all the courage in the world."