In the past few years, seismic tomography has begun to provide detailed images of seismic velocity in the Earth's interior which, for the first time, give direct observational constraints on the mechanisms of heat and mass transfer. The study of surface waves has led to quite detailed maps of upper-mantle structure, and the current global models agree reasonably well down to wavelengths of approximately 2000km. Usually, the models contain only elastic isotropic structure, which provides an excellent fit to the data in most cases. For example, the variance reduction for minor and major arc phase data in the frequency range 7-15mHz is typically 65-92 per cent and the data are fit to within 1-2 standard deviations. The fit to great-circle phase data, which are not subject to bias from unknown source or instrument effects, is even better. However, there is clear evidence for seismic anisotropy in various places on the globe. This study demonstrates how much (or little) the fit to the data is improved by including anisotropy in the modelling process. It also illuminates some of the trade-offs between isotropic and anisotropic structure and gives an estimate of how much bias is introduced by neglecting anisotropy. Finally, we show that the addition of polarization data has the potential for improving recovery of anisotropic structure by diminishing the trade-offs between isotropic and anisotropic effects.