The linothorax was a type of armour popular in classical Greece and the Hellenistic empires that succeeded it. So, from at least as early as the 7th Century BC, and in use until about the 1st Century BC. It generally consisted of a cuirass and 'skirt' of pteruges and is thought to have been made of repeated layers of linen1 cloth, stiffened with resin, glue or wax.
The result was normally a relatively light, cool - and most importantly cheap - form of armour which was used extensively in its home culture, mostly for cavalry and light infantry such as the peltast, although it was frequently issued to pikemen and Hoplites as well.
Linothorax is a virtual unknown even to most gamers, as it appears almost nowhere in popular culture. Its existence is not all that widely remembered and there are no surviving examples2. Just one possible half-rotted partial remnant3 was found in a tomb. However, images of the linothorax adorn numerous pottery vessels from Ancient Greece, and there's a famous mosaic commemorating the victories of Alexander the Great that shows him wearing a linothorax in combat. Several literary references exist as well, ranging from Homer to Herodotus.
Some depictions of linothorax in classical art show sections that have been reinforced with plates or scales of metal. Especially around the belly, the most likely target area given the under-hand thrusting style of the Greek Hoplites in warfare. Thus, linothorax could be (if the GM were so inclined) further defined into "heavy" and "light" versions, with or without metal reinforcement, respectively. Alternatively the linothorax with metal scales attached could simply be classified as scale mail.
Recently, a team at University of Wisconsin - Green Bay has made suits of linothorax using traditional period-authentic supplies of linen and a glue made from rendering rabbits.  The resulting armor is very resistant to arrows as well as blows with various melee weapons. 
Game and Story Use
- Obviously important if the campaign takes place in historical Ancient Greece or a fantasy counterpart culture thereof, but could also be produced by pretty much any other culture, especially ones that operate in a hot environment.
- Linothorax would prove extremely advantageous to a quickly-mustered army fighting in warm climates. It's light weight, and doesn't heat up under the hot sun like bronze does. As it is worn, the heat and moisture of your body causes it to conform to your shape, reducing fatigue and movement restrictions. This means it can be made in bulk, and "broken in" during deployment, whereas most metal armors need a greater degree of custom-fitting.
- Obviously this design bears more than a passing resemblance to modern resin-bonded kevlar armours - that's probably just a coincidence, but could make good time traveller tech…