  We use a high-resolution cosmological simulation to predict the distribution
of HI Ly-alpha emission from the low-redshift (z<0.5) intergalactic medium
(IGM). Our simulation can be used to reliably compute the emission from
optically thin regions of the IGM but not that of self-shielded gas. We
therefore consider several models that bracket the expected emission from
self-shielded regions. Most galaxies are surrounded by extended (>10^2 kpc)
``coronae'' of optically thin gas with Ly-alpha surface brightness close to the
expected background. Most of these regions contain smaller cores of dense, cool
gas. Unless self-shielded gas is able to cool to T<10^4.1 K, these cores are
much brighter than the background. The Ly-alpha coronae represent ``cooling
flows'' of IGM gas accreting onto galaxies. We also estimate the number of
Ly-alpha photons produced through the reprocessing of stellar ionizing
radiation in the interstellar medium of galaxies; while this mechanism is
responsible for the brightest Ly-alpha emission, it occurs on small physical
scales and can be separated using high-resolution observations. In all cases,
we find that Ly-alpha emitters are numerous (with a space density ~0.1 h^3
Mpc^-3) and closely trace the filamentary structure of the IGM, providing a new
way to map gas inside the cosmic web.
