Growth of nm-thick yttrium iron garnet (YIG) films by sputtering and ferromagnetic resonance (FMR) properties in the films were studied. The FMR linewidth of the YIG film decreased as the film thickness was increased from several nanometers to about 100 nm. For films with very smooth surfaces, the linewidth increased linearly with frequency. In contrast, for films with big grains on the surface, the linewidth-frequency response was strongly nonlinear. Films in the 7-26 nm thickness range showed a surface roughness between 0.1 nm and 0.4 nm, a 9.48-GHz FMR linewidth in the 6-10 Oe range, and a damping constant of about 0.001.