An insulating liner for use with exhaust system or pollution control devices such as catalytic converters and diesel particulate filters or traps. The insulating liner is shown in relation to an end cone for use with a catalytic converter. The end cone includes an outer metallic end cone and a free-standing insulating cone positioned within the outer metallic end cone. A substantial portion of the inner surface of the insulating liner is exposed to hot exhaust gas from an internal combustion engine. The insulating liner is preferably formed of a composite containing inorganic fibers and/or particles, which makes the insulating liner rigid, yet capable of withstanding repeated mechanical and thermal shocks.