We simulate global seismic wave propagation based upon a spectral-element method. We include the full complexity of 3-D Earth models, i.e. lateral variations in compressional-wave velocity, shear-wave velocity and density, a 3-D crustal model, ellipticity, as well as topography and bathymetry. We also include the effects of the oceans, rotation and self-gravitation in the context of the Cowling approximation. For the oceans we introduce a formulation based upon an equivalent load in which the oceans do not need to be meshed explicitly. Some of these effects, which are often considered negligible in global seismology, can in fact play a significant role for certain source–receiver configurations. Anisotropy and attenuation, which were introduced and validated in a previous paper, are also incorporated in this study. The complex phenomena that are taken into account are introduced in such a way that we preserve the main advantages of the spectral-element method, which are an exactly diagonal mass matrix and very high computational efficiency on parallel computers. For self-gravitation and the oceans we benchmark spectral-element synthetic seismograms against normal-mode synthetics for the spherically symmetric reference model PREM. The two methods are in excellent agreement for all body- and surface-wave arrivals with periods greater than about 20 s in the case of self-gravitation and 25 s in the case of the oceans. At long periods the effect of gravity on multiorbit surface waves up to R4 is correctly reproduced. We subsequently present results of simulations for two real earthquakes in fully 3-D Earth models for which the fit to the data is significantly improved compared with classical normal-mode calculations based upon PREM. For example, we show that for trans-Pacific paths the Rayleigh wave can arrive more than a minute earlier than in PREM, and that the Love wave is much shorter in duration.