A large data set of fundamental mode Rayleigh wave amplitudes is analyzed to derive global models of surface wave attenuation (1/Q). The data set consists of measurements of Rayleigh wave amplitude anomalies in the period range 50--250 s for 347 earthquakes observed at 179 seismic stations. The amplitude anomalies are considered to depend on four factors: intrinsic attenuation along the ray path, elastic focusing effects along the ray path, a source factor accounting for uncertainties in the strength of excitation, and a receiver factor accounting for uncertainties in the response at the station. The amplitude data are inverted simultaneously for global maps of attenuation expanded in spherical harmonics up to degree 12, global maps of phase velocity expanded to degree 20, and source and receiver correction factors. All four variable types are shown to be important in explaining the amplitude anomalies. A data set of phase delay measurements provides additional constraints on velocity structure. The maps of attenuation obtained by simultaneous inversion for elastic and anelastic models contain important features that are not robustly imaged when the effect of focusing on wave amplitude is ignored. These include high attenuation along western North America and along the East Pacific Rise and other ridge systems and low attenuation associated with stable continental interiors. The global attenuation maps exhibit a strong correlation with phase velocity maps corrected for the effect of the crust, particularly for periods <200 s. The correlation suggests that the variability in both Q and velocity in the shallow upper mantle is primarily thermal in origin.