Global Positioning System (GPS) data from campaigns carried out over the 5 years following the 1989 Loma Prieta earthquake and leveling data measured in 1990 and 1992 define the postseismic velocity field around the Loma Prieta rupture zone. Subtraction of a background velocity field yields a residual velocity pattern which we interpret as the product of two physical processes: (1) slow afterslip along distinct planes in the upper crust and (2) viscoelastic relaxation of the lower crust and upper mantle. B¿rgmann et al. <1997> previously derived an afterslip model involving uniform afterslip on two optimally determined planes, including oblique reverse slip on the coseismic rupture and reverse slip on a shallow thrust fault to the northeast of the San Andreas fault. We further consider models of distributed slip on these two fault planes plus a viscoelastic relaxation pattern which depends on a suitable coseismic rupture model and crust and mantle viscosities. Several fault models from the literature were considered for the 1989 coseismic rupture, with nearly identical impact on the results. Simultaneous maximum likelihood inversion of the GPS and leveling data for afterslip distribution and viscosity yields the following results: (1) A good fit to the data is obtained by smooth afterslip distributions without any viscoelastic relaxation being required. (2) Tangible broad-scale viscoelastic relaxation of the lower crust and upper mantle are present in this data set at 97% confidence, and a lower crustal viscosity of ~1019 Pa s is obtained; however, the viscosity of both the lower crust and upper mantle are poorly constrained. (3) For a given misfit, 20% less integrated afterslip and smoother afterslip distributions result when viscoelastic relaxation is included. (4) Maximum slip rates on the slip-distributed models are 3--5 cm/yr, the dominant patches estimated on the two planes fill in the entire depth range 4--13 km without significant overlap, and deeper afterslip is not required. The afterslip distribution on the coseismic rupture plane is strongly dominated by reverse slip immediately southeast of the main center of coseismic reverse slip. ¿ 1998 American Geophysical Union