The recent deep X-ray surveys suggest that discrete sources comprise most, if not all, of the energy content of the background (cf. Giacconi, this volume). We have shown that, if sources have flat power law spectra and relatively sharp high-energy cutoffs, their combined emission can also mimic very accurately its extended (2 - 400 keV) spectral shape. A broad distribution of cutoff energies Ec is required, in this case, to model the high-energy (E ≳ 20 - 40 keV) part: a power law envelope of distribution functions of various types of sources can be envisaged; alternatively, the well-known fact that most of the 2 − 10 keV background should be produced by low flux, low Ec sources can suggest that Ec ∝ (1 + z)−η. The quality of the fit turns out to be not very sensitive to the amount of number/luminosity evolution assumed, though the minimum χ2 test slightly favours strongly evolving sources. In the case of differential luminosity ∝ E−γ and evolution ∝ (1 + z)α with 0 ≤ α ≤ 6, the best fits to pre HEAO-1 data are obtained for mean values of the spectral index γ ≈ 0.5 − 0.9 and for dispersions Δγ ≃ 0.5 - 0.7. Rather wide ranges of values of γ and Δγ are, however, still allowed; e.g., for α = 6, the allowed intervals are 0.2 ≤ γ ≤ 1.3, 0 ≤ Δγ ≤ 0.7.