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Bow and arrow are one of the oldest tools of hunt and war. They were known to almost all civilizations since before the written history and are still in use today although not in such extent. Read about history of bow and arrow.
Crossbow is a younger sibling of a bow and arrow. It is easier to use and more precise with less effort but slower. That is why it replaced bow and arrow in battlefields until it was itself replaced with firearms.
Composite bow, although a bow if a pretty complex design, is old weapon and many civilizations knew about it and used it above others. It can be made to be much smaller and stronger but it could last for a shorter time if exposed to humidity.
Compound bow is one of the most modern designs of a bow. It uses more complex physics to give more power to the arrow and allows an archer to hold the arrow in full draw without too much strength.