The arrival times of direct P and S waves, measured on seismograms recorded from natural lunar seismic events, have been analyzed using linearized inversion and parameter search methods to simultaneously determine event locations, origin times, and structural parameters (seismic velocities). Polarization-filtered record section plots are correlated with theoretical travel time curves to identify later phases and obtain additional structural and velocity information. Shear wave amplitudes plotted as a function of distance provide data on the existence and magnitude of seismic velocity gradients in the interior. These studies are used to delineate the structure of the moon below the crust to a depth of about 1100 km. This structure has been divided into an upper and a lower mantle. The upper mantle, from 60- to 400-km depth, has average seismic wave velocities of Vp = 7.70¿0.15 km/s and Vs = 4.45¿0.05 km/s with corresponding Q values (taken from the literature) of Qp~5000 and Qs~3000. The shear wave velocity decreases with a negative gradient of at most -6¿10-4 km/s/km, implying a Vs variation of from 4.57 km/s at 60-km depth to 4.37 km/s at 400-km depth; this decrease can be accounted for by increasing temperature, and so no major compositional gradient is required in the upper mantle. A small negative P wave velocity gradient may also be present. Between 400- and 480-km depth there is a transition zone with a sharply decreasing shear wave velocity and possibly an accompanying small decrease in Vp. The dominant shear velocity decrease may occur at a 480-km interface and may represent a compositional change, although the effects of increased temperature cannot be totally ruled out. The lower mantle extends from 480-km depth to at least 1100-km depth; few seismic waves recorded by the lunar network penetrate below 1100 km. The average velocities are Vp = 7.6¿0.6 km/s and Vs = 4.2¿0.1 km/s. Q values (taken from the literature) are Qp~1500 and Qs~1000. Below 1100 km there is some indication that the attenuation may increase still further for shear waves, with Qs dropping to a few hundreds or less. This seismic model of the moon is well constrained, with uncertainties on the above values given explicitly by the analysis methods, and so it serves as a strong control on the present-day lunar compositional and thermal structure.