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  There are two main surround types used today, half-roll and accordian.
Half-roll (positive, negative) is a commony used in home speakers, but high excursion can pull the cone to one side.
Accordion surrounds are generally made of cloth, and are used in high-powered audio setups. Accordion surrounds can move more than half-roll, but at the surround resonance dips in frequency response can occour.
Butyl rubber and foam surrounds are used to aleviate some of the problems of paper surrounds. Foam surrounds do disentagrate after about 30 years, but they are light and allow high excursion. Rubber surrounds are heavy which reduces effencency, but they do not disentagrate. On a sidenote, the Qms is generally a measure of friction losses from the suspension, with values from 1-5 indicating a surround that will control a cone well.

The surround, which acts as an air seal between the cone and the basket, adds to the restoring force of the spider. Another function of the surround, is to absorb cone flexure waves as they are transferred up the cone.

At very thin surround sound can penetrate through it and partially cancel and color of direct radiation sound.