We use a high-resolution cosmological simulation to predict the distribution of H I Lyalpha emission from the low-redshift (zless than or similar to0.5) 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 (greater than or similar to10(2) kpc) "coronae" of optically thin gas with Lyalpha 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(41), these cores are much brighter than the background. The Lyα coronae represent "cooling flows" of IGM gas accreting onto galaxies. We also estimate the number of Lyα photons produced through the reprocessing of stellar ionizing radiation in the interstellar medium of galaxies; while this mechanism is responsible for the brightest Lya emission, it occurs on small physical scales and can be separated using high-resolution observations. In all cases, we find that Lyα emitters are numerous ( with a space density of ∼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.