Armstrong, the American astronaut who made "one giant leap for mankind" when he became the first man to walk on the moon, died Saturday.
"We are heartbroken to share the news that Neil Armstrong has passed away following complications resulting from cardiovascular procedures," Armstrong's family said in a statement.
Armstrong underwent heart surgery earlier this month.
"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.
NASA administrator Charles Bolden put Armstrong's death in perspective.
"As long as there are history books, Neil Armstrong will be included in them," said Bolden.
"As we enter this 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."
Armstrong took two trips into space. He made his first journey in 1966 as commander of the Gemini 8 mission, which nearly ended in disaster.
Armstrong kept his cool and brought the spacecraft home safely after a thruster rocket malfunctioned and caused it to spin wildly out of control.
During his next space trip in July 1969, Armstrong and fellow astronauts Buzz Aldrin and Michael Collins blasted off in Apollo 11 on a nearly 250,000-mile journey to the moon that went down in the history books.
It took them four days to reach their destination.
The world watched and waited as the lunar module "Eagle" separated from the command module and began its descent.
Then came the words from Armstrong: "Tranquility Base here, the Eagle has landed."
About six and a half hours later at 10:56 p.m. ET on July 20, 1969, Armstrong, at age 38, became the first person to set foot on the moon.
He uttered the now-famous phrase: "That's one small step for (a) man, one giant leap for mankind."
The quote was originally recorded without the "a," which was picked up by voice recognition software many years later.
Armstrong was on the moon's surface for two hours and 32 minutes and Aldrin, who followed him, spent about 15 minutes less than that.