Armstrong died just weeks after he underwent a heart surgery, and his family said in a statement that he passed away following complications resulting from cardiovascular procedures, according to United States media reports.
July 20, 1969, became a watershed date in the history of mankind after Armstrong, commanding the Apollo 11 spacecraft landed on the moon, and accompanied by fellow astronaut Edwin Aldrin spent nearly three hours walking on the moon.
"That's one small step for a man, one giant leap for mankind," is how Armstrong had broken the historic news from the satellite when he radioed back.
It had taken Apollo 11 craft, carrying Armstrong and fellow astronauts Buzz Aldrin and Michael Collins, four days to complete the nearly 250,000-mile (400,000 km) journey, as the world waited with bated breath.
"While we mourn the loss of a very good man, we also celebrate his remarkable life and hope that it serves as an example to young people around the world to work hard to make their dreams come true, to be willing to explore and push the limits, and to selflessly serve a cause greater than themselves," his family said in a statement.
"For those who may ask what they can do to honour Neil, we have a simple request. Honour his example of service, accomplishment and modesty, and the next time you walk outside on a clear night and see the moon smiling down at you, think of Neil Armstrong and give him a wink," they said.
Armstrong was 38 years old when he became the first man to set foot on the moon.
The development that came at a time of intense space rivalry between the reigning superpowers -- the US and the USSR -- gave America a major leap over its Cold War foe.
As soon as the news of Armstrong's death spread, tributes started pouring in for the legendary former astronaut.
"Neil Armstrong was a true American hero, both because of his extraordinary service to his country and the honourable life he led. He was a groundbreaking naval aviator and the world's most famous astronaut, but it was his humble and gracious response to the torrent of attention that followed his accomplishments that may have set him apart most," said Ohio Senator Rob Portman.
It was in Ohio town of Wapakoneta that Armstrong was born on August 5, 1930.
Later on, Armstrong chose to avoid the public limelight and lead an intensely private life. He resigned from NASA in 1971 to take up teaching engineering at the University of Cincinnati for several years.
In fact, his family in the statement desribed him as "a reluctant American hero who always believed he was just doing his job."
In one of his rare public appearances at a gathering with Aldrin and other Apollo astronauts to mark the 30th anniversary of their moon landing, he recognised the importance of his achievement.
"In my own view, the important achievement of Apollo was a demonstration that humanity is not forever chained to this planet, and our visions go rather further than that, and our opportunities are unlimited," Armstrong said.