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US President John F. Kennedy made that declaration at Rice Stadium, in Texas, in 1962. He would not live to see the dramatic moment Neil Armstrong set foot on the moon on July 21, 1969, but he would have approved, not only because it meant the Americans had won the "space race" with the Soviet Union but because it was a moment that united the world in wonder.
It was a "giant leap for mankind", a triumph for technology, and a moment that still resonates, 50 years after the crew of Apollo 11 completed the first successful moon landing by a manned craft.
The numbers are staggering. Nasa spent $US25billion - about $NZ230billion in 2019 - on the Apollo programme. More than 500million watched the moon landing on television. Some 400,000 people worked on the mission.
Beyond the figures and the advances in science that allowed it to happen, the landing of humans on the moon speaks to our instinct to explore and our interest in the universe around us, and is a reminder of mankind's ability to achieve wonderful things.
The moon is our constant companion, like the sun's nocturnal little brother. It has beckoned to us, inspired us, fascinated us for centuries.
Did we conquer it? No. But we travelled to it, and we walked on it, and that is an amazing feat matched by few others.
We are an odd species. Humans can be cruel and destructive and short-sighted. They can make things like nuclear bombs with one sole purpose. Yet they can also be compassionate, kind and smart, and they can harness their knowledge and skills to get human beings safely to the moon and back.
The spirit and science that got us to the moon could be required again as humanity faces arguably its greatest current challenge, the rising threat of climate change. Mankind is capable of destroying the world. It is not a giant leap to suggest that means we are capable of saving it.
What, then, of the moon?
Many are surprised to learn there has not been a human foot on the lunar surface since the Apollo 17 mission in 1972, when American astronaut Eugene Cernan was the last man off. It has only since been visited by unmanned vehicles. The feeling, clearly, was that as the Americans had won the race to the moon, there was little point in pushing for second place.
There is something of a new space race happening, however. China has led the way - landing the Chang'e-3 probe on the moon in 2013, and showing a clear intention to get people there in the future. India is just about to send its own probe to look for water molecules. And the US Administration is in a mood for the moon again, promising to return astronauts to the land of green cheese by 2024.
To some, the moon is yesterday's news. We've been, we've seen, we've got the T-shirt. Pushing further - the ultimate goal being to land a human on Mars, and to explore the possibilities for civilisation there and beyond - is a much sexier prospect.
But the moon will always be special. Getting there was the ultimate adventure, and is as much a marvel today as it was 50 years ago.