I present the results of a number of simulations of the dynamical evolution of globular cluster systems (GCS) in elliptical galaxies including the effects of two-body relaxation, dynamical friction, stellar evolution and the effects of the tidal field of the host galaxy. The results of detailed models for the evolution of the M87 GCS are also presented. A significant number of clusters are disrupted by evolutionary processes and the properties of many of the clusters which survive are effected by dynamical evolution. In spite of large differences in the efficiency of evolutionary processes in different galaxies, the final galaxy-to-galaxy variation of the GCS mean mass and its radial variation within individual galaxies in my simulations are small and consistent with observations. The effects of dissolution of low-concentration clusters due to mass loss through stellar evolution are also discussed and are shown to play an important role in the evolution of a power-law GCS mass function (GCMF), similar to that observed in young cluster systems in merging galaxies, towards a final GCMF with properties consistent with observations.