We prove that amongst all fullerenes the dodecahedron has maximum smallest eigenvalue (equal to -), followed by the three fullerenes that have all their hexagons disjoint (the unique fullerenes on 24 and 26 vertices, and the tetrahedral fullerene on 28 vertices), for which the smallest eigenvalue is in each case equal to - 1 -. We also prove that amongst all IPR fullerenes the icosahedral C₆₀ fullerene has maximum smallest eigenvalue (equal to ² where is the golden ratio (1 +/2).