The American astronaut had been hospitalised recently with heart surgery and his family said in a statement that he had died following complications resulting from cardiovascular procedures.
Armstrong became famous around the world in 1969 with the Apollo 11 mission to the moon where he was joined on the mission by Buzz Aldrin and Michael Collins.
In his greatest act Armstrong became the first man to walk on the moon on July 21, 1969 when he stepped off the lunar lander and said "That's one small step for man, one giant leap for mankind."
NASA Administrator Charles Bolden said: "On behalf of the entire NASA family, I would like to express my deepest condolences to Carol and the rest of the Armstrong family on the passing of Neil Armstrong. As long as there are history books, Neil Armstrong will be included in them, remembered for taking humankind's first step on a world beyond our own."
Besides being one of America's greatest explorers, Neil carried himself with a grace and humanity that was an example to us all. When President Kennedy challenged the nation to send a human to the moon, Neil Armstrong accepted without reservation.
"As we enter the next era of space exploration, we do so standing on the shoulders of Neil Armstrong. We mourn the passing of a friend, fellow astronaut and true American hero."
During his lifetime, Armstrong was awarded with the Presidential Medal of Freedom and the Congressional Space Medal of Honour along with the historical status of being the first man to walk on the moon.
