A fullerene graph (in short a fullerene) is a 3-connected cubic planar graph all of whose faces are pentagons and hexagons. A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. The existence of semiregular automorphisms in fullerenes is discussed. In particular, the family of fullerene graphs is described via the existence of semiregular automorphisms in their automorphism groups.