If the luminosity per mass of the universe at redshifts z = 5-1000 were at least comparable to its present luminosity, then a conspicuous cosmological infrared radiation background would be produced. A number of situations where this could arise are surveyed, and the intensity of the background is evaluated for specific types of sources (protogalaxies, pregalactic stars, quasars, black holes, and decaying relict particles) in several candidate scenarios, which are also discussed in terms of metal enrichment, dark matter, and formation of large-scale structure. The spectrum of the background radiation is estimated, both with and without dust obscuration. General features of cosmological radiative transfer with dust are discussed. It is argued that dust is expected to degrade the background to the far-infrared, 100-1000 microns, where the wavelength of the spectral peak can be predicted from the total present-day background flux and depends only weakly on properties of the dust or the redshift of emission. The statistical properties of the anisotropy expected in the radiation and its relationship to the distribution of dust at the time the dust is formed or the radiation is produced are estimated. Intensity fluctuations at the few percent level on arc minute scales are typical inmore » current galaxy-formation scenarios. 84 references.« less