Rest-frame UV emission lines offer the possibility to directly image the gas around high-redshift galaxies with upcoming optical instruments. We use a suite of large, hydrodynamical simulations to predict the nature and detectability of emission lines from the intergalactic medium at 2<z<5. The brightest emission comes from HI Ly-alpha and the strongest metal line, CIII, is about an order of magnitude fainter, although HI Ly-alpha may be fainter if the gas is self-shielded to the UV background or if dust is important. The highest surface brightness regions for CIV, SiIII, SiIV and OVI are fainter than CIII by factors of a few. The NV and NeVIII lines, as well as HeII H-alpha, are substantially weaker but their maximum surface brightnesses still exceed 100 photon/cm^2/s/sr at z=2 (for 2" pixels). Lower ionisation lines arise in denser and colder gas that produces clumpier emission. The brightest HI Ly-alpha emission arises in highly overdense gas, but the highest surface brightness emission from high-ionisation metal lines traces a wider range of overdensities. Bright metal-line emission traces gas with temperatures close to the peak of the corresponding emissivity curve. While HI Ly-alpha, HeII H-alpha, CIII, SiIII, and SiIV are excellent probes of cold accretion flows and the colder parts of outflows, CIV, NV, OVI, and NeVIII are powerful tracers of the diffuse WHIM and galactic winds. A comparison of results from simulations with varying physical prescriptions demonstrates that the predictions for the brighter metal-line emission are robust to within factors of a few. Several emission lines from the high-redshift IGM will become detectable in the near future, possibly starting with the Cosmic Web Imager on Palomar. MUSE and the Keck Cosmic Web Imager have the potential to revolutionise studies of the interactions between high-redshift galaxies and their environment. (Abridged)