Distance matrices of fullerenes which contain important topological information on fullerenes are obtained. By use of the distance matrices, the distance level patterns are constructed for fullerenes. It is shown that the distance level patterns uniquely characterize fullerenes and thus appear to be topological invariants of fullerenes. The distance spectra and characteristic polynomials of the distance matrices of C-20-C-90 fullerenes are computed. The Wiener topological indices are computed for these fullerenes. The sums of powers of the distance spectra are computed and considered as structural invariants of fullerenes.