  Radiation in the extreme ultraviolet (EUV) and soft X-ray holds clues to the
location of the 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 O(1) 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.
