His family says he died from complications from heart surgery he had earlier this month.
Neil Armstrong walked on the Moon on 20 July 1969, famously describing the event as “one small step for [a] man, one giant leap for mankind”.
Last November Neil Armstrong received the Congressional Gold Medal, the highest US civilian award.
Neil Armstrong was the commander of the Apollo 11 spacecraft. He and fellow astronaut Edwin “Buzz” Aldrin spent nearly three hours walking on the moon.
The mission was his last space flight. He left the US space agency NASA to teach aeronautics in 1971.
Born in 1930 and raised in Ohio, Neil Armstrong took his first flight aged six with his father and formed a lifelong passion for flying.
He flew Navy fighter jets during the Korean War in the 1950s, and joined the US space programme in 1962.
Correspondents say Neil Armstrong remained modest and never allowed himself to be caught up in the glamour of space exploration.
“I am, and ever will be, a white-socks, pocket-protector, nerdy engineer,” he said in February 2000, in a rare public appearance.
In a statement, his family praised him as a “reluctant American hero” who had “served his nation proudly, as a navy fighter pilot, test pilot, and astronaut”.
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