A great variety of materials has been used from the XIXth century to our present days.
As natural materials we used Cotton, wool, silk, linen and even straw. Then artificial materials came to appear, as for example rayon (artificial silk) and acrylic.
Nylon and polyester were also used as synthetic materials.
If rubber or some elastic material is put inside a braid, it becomes elastic as a whole (as bungees are made, for example).