The band gap energy was estimated experimentally by measuring the change in optical absorption intensity as a function of photon energy in a wide range from visible to vacuum-ultraviolet wavelengths for commercially available 11 kinds of insulating polymers. The absorption spectra as a function of photon energy measured by synchrotron radiation and by a spectrometer agree with each other at the photon energies where the two measurements were available. The band gap energy is quite high in linear polyolefin polymers. Namely, it is 6.9 and 7.0 eV for polyethylene and polypropylene, respectively. In these two polymers, the relative permittivity at 1 kHz is quite low in both polymers. On the other hand, the polymers with aromatic rings in their structures have low band gap energies and large permittivities. Namely, a negative relationship between the band gap energy and permittivity, which is well known for inorganic solids, is also seen in insulating polymers. Furthermore, to investigate the possibility of appearance of midgap localized states by the addition of nanofillers, absorption and luminescence spectra were examined for polymer nanocomposites based on polyethylene and polyamide. As a result, the nanofiller addition was found to give no influences on their band gaps.