The electronic energy-band structure of tungsten has been calculated by means of the relativistic-augmented-plane-wave method. A series of mutually related potentials are constructed by varying the electronic configuration and the amount of Slater exchange included. The best band structure is obtained from an ad hoc potential based on a Dirac-Slater atomic calculation for the ground-state configuration and with full Slater exchange in the atomic as well as in the crystal potential. The selection of this best potential is justified by comparing the calculated band structure to Fermi-surface experiments and to optical-reflectance measurements up to 5-eV photon energy. The temperature and strain responses in the band structure are estimated from band calculations with four different lattice constants. The band structure was determined in the entire Brillouin zone and is applied to a calculation of photoemission spectra from W single crystals. The nondirect as well as the direct models for bulk photoemission processes are investigated. The emission from the three low-index surfaces (100), (110), and (111) exhibits strong dependence on direction and acceptance cone. According to the present band model there should essentially be no emission normal to the (110) face for photon energies between 9.4 and 10.6 eV. Experimental observation of emission in this gap, however, implies effects not included in the simple bulk models. In particular, effects arising from surface emission have been considered, i.e., emission of those electrons which are excited in a single-step process from initial states near the surface to final states outside the crystal. The electrons that are emitted from the surface in directions perpendicular to the crystal planes carry information on the one-dimensional surface density of states. The present work includes a crude estimate of this surface density of states, which is derived from the bulk band structure by narrowing the d bands according to an effective number of neighbors per surface atom. Estimates of surface relaxation effects are also included.