Armstrong had heart surgery several weeks ago, and a statement from his family said he died following complications resulting from cardiovascular procedures.
"Neil Armstrong was also a reluctant American hero who always believed he was just doing his job," his family said. "He served his Nation proudly, as a navy fighter pilot, test pilot, and astronaut. ... He remained an advocate of aviation and exploration throughout his life and never lost his boyhood wonder of these pursuits."
Read the full statement from Neil Armstrong's Family
On July 20, 1969, half a billion people -- a sixth of the world's population at the time -- watched a ghostly black-and-white television image as Armstrong backed down the ladder of the lunar landing ship Eagle, planted his left foot on the moon's surface, and said, "That's one small step for (a) man, one giant leap for mankind."
Twenty minutes later his crewmate, Buzz Aldrin, joined him, and the world watched as the men spent the next two hours bounding around in the moon's light gravity, taking rock samples, setting up experiments, and taking now-iconic photographs. The third member of their crew, Michael Collins, orbited overhead in the Apollo 11 command ship, Columbia.
"Neil and I trained together as technical partners but were also good friends who will always be connected through our participation in the mission of Apollo 11," said Aldrin today in a statement. "Virtually the entire world took that memorable journey with us. I know I am joined by millions of others in mourning the passing of a true American hero and the best pilot I ever knew."
Collins said, "He was the best, and I will miss him terribly."
President Obama issued a statement from the White House: "Neil was among the greatest of American heroes -- not just of his time, but of all time," it said. Armstrong and his crewmates "set out to show the world that the American spirit can see beyond what seems unimaginable -- that with enough drive and ingenuity, anything is possible."
NASA Administrator Charles Bolden -- himself a former space shuttle astronaut -- joined in the tributes. With the space shuttles retired, NASA does not currently have a way to launch astronauts on its own, but it is working on a new spacecraft, and, this month, landed the robotic Curiosity rover on Mars.
"Besides being one of America's greatest explorers, Neil carried himself with a grace and humility that was an example to us all," said Bolden. "As we enter this next era of space exploration, we do so standing on the shoulders of Neil Armstrong."
Armstrong's step fulfilled a challenge laid down by an earlier president, John F. Kennedy, in May 1961. Struggling in his first months in the White House, Kennedy addressed a joint session of Congress:
"I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the earth," he said. "No single space project in this period will be more impressive to mankind, or more important for the long-range exploration of space; and none will be so difficult or expensive to accomplish."
Armstrong was a 30-year-old test pilot at the time of Kennedy's challenge, flying the X-15 rocket plane for a new government agency called NASA. He had served as a Naval aviator in the Korean War, flying 78 missions, and had an engineering degree from Purdue University. A native of the small town of Wapakoneta, Ohio, he was married to the former Jan Shearon and living near Edwards Air Force Base in the high desert of California.
NASA already had seven astronauts, flying its Mercury space capsule. In 1962 it sent out word that it was looking for more, and Armstrong was one of the nine it selected.
On March 16, 1966 he became the first American civilian to orbit the earth, commanding the two-man Gemini VIII mission with David R. Scott as his crewmate. On their fourth orbit, they made the first-ever docking in space with another spacecraft -- a maneuver the still-untested Apollo project would need to get astronauts to and from the lunar surface.
Minutes later, though, the spacecraft began to tumble wildly out of control, apparently because of a broken maneuvering thruster. It was a dangerous moment -- a 6,000-pound ship, moving at 17,500 mph, spinning and turning end-over-end once a second. Armstrong ended the emergency by using a second set of thrusters. Mission Control ordered the astronauts to land as soon as possible, and after 10 hours of flight they splashed down safely in the Pacific.
The two astronauts were commended for keeping their cool in a difficult situation, and when Project Apollo began, Armstrong was assigned to command one of the first six flights. At the time this was not momentous news. NASA had a system for rotating its crews among flights -- one served as backup crew for a mission and then actually flew three flights later -- and nobody knew how many test flights would be needed before the first moon landing could be attempted.