Bow and Arrow
(Really shooting an arrow, but you can see it really well here)
People probably invented bows and arrows about 64,000 years ago. Some of the earliest arrowheads come from South Africa. As people spread from Africa to India, Australia, all over Asia, and Europe, they took their bows and arrows with them. People in Australia seem to have forgotten about the bow and arrow, but people did use bows and arrows all over Asia, Africa, and Europe.
A bow and arrow is a kind of lever, where your hand on the bow acts as the fulcrum for the lever. The arrows are wooden sticks fletched with feathers stuck to the ends to balance the arrows and help them to fly in a straight line.
To make a strong bow out of wood, it has to be pretty long - nearly as tall as a person. When people began to ride horses in Central Asia, about 2500 BC, these long bows got in their way. So about this time, somebody in Central Asia invented the stronger, more flexible composite bow. The composite bow is made of layers of wood, animal horns, and sinew, glued together in layers (usually a layer of sinew, then wood in the middle, then horn on the other side), and you can bend it a lot farther without breaking it, so you can shoot arrows with a shorter bow that will fit better on your horse. Gradually the invention spread: by 1300 BC, Egyptian soldiers were also riding horses and using composite bows, and by about 300 BC Chinese soldiers were using composite bows.
Persian archers with recurve bows (Susa, 500 BC)
Later on, maybe around 1000 BC, these same horse riding archers in Central Asia invented the recurve bow. These bows were in the shape of a wide W, which can shoot further for the same length of bow. The use of recurve bows quickly spread from Central Asia to China. The Persians may have brought the recurve bow to West Asia, and from there to the Egyptians, Phoenicians and Greeks, who spread it to the Romans and Carthaginians in Africa.
Soon after the recurve bow came to China, about 450 BC, Chinese blacksmiths invented the crossbow. Crossbows, which had an iron catch added, allowed you to draw the bow, getting it all ready to shoot, and then stop it there, so you could shoot just by releasing the catch. A mechanical crank let you draw the bow much harder than your muscles could alone, which allowed you to shoot much farther than with a regular bow. Roman hunters had learned about crossbows by around 50 AD, and soldiers in the Abbasid empire were using crossbows before 900 AD.
In the Middle Ages, many archers in Europe used crossbows. Even in the Middle Ages, though, archers in England, France, and Germany often used all-wood longbows instead of recurve composite bows or crossbows, because they lasted better in wet weather, and these archers weren't riding horses anyway.
Even though bows and arrows already existed by the time people first crossed over to North America, those people don't seem to have known about them: the earliest people living in North America didn't use bows and arrows. But around 500 AD, people in North America invented the bow and arrow for themselves. Especially on the West Coast, many people also used recurve bows. Because people in North America didn't ride horses, they didn't need composite bows, but some bow-makers did still use sinew on their bows, and as soon as North American people got horses from the Spanish invaders in the 1500s, they began to invent shorter composite bows.