Historically accurate, this weave has been used as far in time as the third and second century BC. It is usually associated with medieval armour, since it fell into disuse between the fall of the Roman empire and the 12th century. There are a number of variation on this pattern, usually named for the number of rings going through each other. The most common is 4 in 1, where, as you can guess, each ring goes through 4.
Historical weave, used it the construction of medieval japanese armours. There is a number of variation on this weave, depending on the exact tessallation of the rings. The most common are Hitoye-Gusari, a square grid, and Hana-Gusari, an hexagonal grid. Hana means "flower" which makes sense when you look at this mail.
Despite its name, there is very little evidence that this chainmail weave comes for Persia. The most common form of this weave is very useful in making chains and bracelets. There are some ways of making an extensive patch of it. The form I usually use is called half-persian 3 in 1, which has interesting asymmetrical look.
Also called Birdcage, so named because of the hollow section which makes this weave distinct. Inserting beads into the cage, or changing the size of the cage allows for this weave to display a versatility in its uses. As with the persian weave, there is very little historical evidence for this weave.
There is a large number of possible weaves. While I have a soft spot for the classical styles, I sometime produce pieces of purely fanciful weaves.