One of the most well known natural polymers is polyisoprene, or natural rubber. Ancient Mayans and Aztecs harvested it from the hevea tree and used it to make waterproof boots and the balls which they used to play a game similar to basketball. It is what we call an elastomer, that is, it recovers its shape after being stretched or deformed. Normally, the natural rubber is treated to give it crosslinks, which makes it an even better elastomer.
Polyisoprene is diene polymer, which is a polymer made from a monomer containing two carbon-carbon double bonds. Like most diene polymers, it has a carbon-carbon double bond in its backbone chain. Polyisoprene can be harvested from the sap of the hevea tree, but it can also be made by Ziegler-Natta polymerization. This is a rare example of a natural polymer that we can make almost as well as nature does.
This is what the isoprene monomer looks like:
Other polymers used as elastomers include: Polybutadiene, Polyisobutylene, Poly(styrene-butadiene-styrene), Polyurethanes, Polychloroprene, Silicones