We have obtained the first average 2-500 keV spectra of Seyfert galaxies, using the data from Ginga and Compton Gamma-Ray Observatory's OSSE. Our sample contains three classes of objects with markedly different spectra: radio-quiet Seyfert 1's and 2's, and radio-loud Seyfert 1's. The average radio-quiet Seyfert 1 spectrum is well-fitted by a power law continuum with the energy spectral index alpha almost-equal-to 0.9, a Compton reflection component corresponding to a approximately 2pi covering solid angle, and ionized absorption. There is a high-energy cutoff in the incident power law continuum: the e-folding energy is E(c) almost-equal-to 0.6(-0.3)+0.8 MeV. The simplest model that describes this spectrum is Comptonization in a relativistic optically-thin thermal corona above the surface of an accretion disk. Radio-quiet Seyfert 2's show strong neutral absorption, and there is an indication that their X-ray power laws are intrinsically harder, although the Seyfert 1 spectrum with alpha = 0.9 and strong reflection cannot be ruled out by the data. Finally, the radio-loud Seyfert spectrum has alpha almost-equal-to 0.7, moderate neutral absorption, E(c) = 0.4(-0.2)+0.7 MeV, and no or little Compton reflection. This is incompatible with the radio-quiet Seyfert 1 spectrum, and probably indicating that the X-rays are beamed away from the accretion disk in these objects. The average spectra of Seyferts integrated over redshift with a power-law evolution can explain the hard X-ray spectrum of the cosmic background. The hump at approximately 30 keV in that spectrum is due to the dominant contribution of Seyfert 2's.