These three peoples lived in a vast area of modern-day Central and South America which incorporates coastal strips, hot and steamy jungles, savannah grassland and cold windy highlands. Though they spoke different languages, they had broadly similar cultures and they worshipped many of the same gods (although they gave them different names). They all used digging sticks, ate maize and beans, respected the number 13 and practised human sacrifice. Interestingly, although they developed the wheel as a toy, for some reason they didn't adapt it for other purposes.
The Aztecs built their settlement in a swamp in what is now Mexico City and when the Spanish arrived they thought it more spectacular than Venice. The Aztecs were fantastic warriors but they were also excellent farmers: because they had stumbled on hydroponics, their floating fields produced an abundance of nutrients in the food they were growing.
The Mayas built some of the tallest buildings of the ancient world – without the use of the wheel, or even horses. The pyramid El Castillo in Chichen Itza is the Mayan calendar, literally set in stone. Each staircase has 91 steps which, when added to the single step at the main entrance to the temple, totals 365 steps. At sunset on the spring equinox, the great serpents' heads at the foot of the main staircase are joined to their tails by a "body" of shadow. They developed a very accurate calendar that could predict solar and lunar eclipses, transits of Venus and - most importantly - the coming of the rains and the time to plant.
These people - known collectively as Mesoamericans - have been processing rubber and latex for over 3500 years (imagine a world without rubber or chewing gum) and they also invented the rubber ball and used it in their deadly ball game where the losing team would be sacrificed.
As an alternative to iron these inventive people made their knives, spears and arrow heads from Obsidian, an extremely sharp stone. Nowadays, surgeons use Obsidian instead of steel scalpels.
They understood the healing power of the rainforest and its plants - amazingly three quarters of all our medicine comes from plants - and in their hands, the wild yam formed the basis of world's first birth control pill. They understood quinine and many European settlers preferred to attend the local Aztec healer rather than their own doctor.
More than half the foods we eat originated in the New World, including potatoes, tomatoes, maize and turkey. But there was one thing that the Aztecs loved above all: chocolate. To them, the cacao bean was so precious they used it as their currency.
Because the Incas lived in the mountains of South America and had to cross very deep ravines they invented the world's first suspension bridges espite not having hemp or sisal ropes. They made their bridges out of twisted fibres of long stemmed grass, another amazing feat of ingenuity from an inspired region.
First broadcast: Wednesday 16 Feb 2005 on BBC TWO