The present invention provides a method for producing a UV induced crosslinked foam in which a melt stream comprising a thermoplastic polymer, a UV crosslinking agent and a photoinitiator is crosslinked by exposing the melt stream to a source of UV light source prior to extruding the melt through an extrusion die. Exposing the melt stream to UV light causes the photoinitiator to become excited and react with the crosslinking agent to generate a free radical. The thus generated free radical reacts with the thermoplastic polymer to induce crosslinking in the polymer. Crosslinking of the polymer can be done prior to extrusion so that the crosslinking process can be carried out without additional steps and/or without the aid of chemical crosslinking agents. The invention also provides an apparatus for producing a crosslinked foam in which the apparatus includes a UV chamber disposed between the extruder and the die through which UV light can be introduced into the melt stream.