Abstract Radiation in the extreme ultraviolet (EUV) and soft X-ray holds clues to the location of missing baryons, the energetics in stellar feedback processes, and the cosmic enrichment history. Additionally, EUV and soft X-ray photons help determine the ionization state of most intergalactic and circumgalactic metals, shaping the rate at which cosmic gas cools. Unfortunately, this band is extremely difficult to probe observationally due to absorption from the Galaxy. In this paper, we model the contributions of various sources to the cosmic EUV and soft X-ray backgrounds. We bracket the contribution from (1) quasars, (2) X-ray binaries, (3) hot interstellar gas, (4) circumgalactic gas, (5) virialized gas, and (6) supersoft sources, developing models that extrapolate into these bands using both empirical and theoretical inputs. While quasars are traditionally assumed to dominate these backgrounds, we discuss the substantial uncertainty in their contribution. Furthermore, we find that hot intrahalo gases likely emit an fraction of this radiation at low redshifts, and that interstellar and circumgalactic emission potentially contribute tens of percent to these backgrounds at all redshifts. We estimate that uncertainties in the angular-averaged background intensity impact the ionization corrections for common circumgalactic and intergalactic metal absorption lines by ≈0.3–1 dex, and we show that local emissions are comparable to the cosmic background only at r prox = 10–100 kpc from Milky Way–like galaxies.