ABSTRACT We present measurements of the luminosity-dependent redshift-space three-point correlation functions (3PCFs) for the Sloan Digital Sky Survey Data Release 7 Main galaxy sample. We compare the 3PCF measurements to the predictions from three different halo and subhalo models. One is the halo occupation distribution (HOD) model and the other two are extensions of the subhalo abundance matching model by allowing the central and satellite galaxies to have different occupation distributions in the host halos and subhalos. Parameters in all the models are chosen to best describe the projected and redshift-space two-point correlation functions (2PCFs) of the same set of galaxies. All three model predictions agree well with the 3PCF measurements for the most luminous galaxy sample, while the HOD model better performs in matching the 3PCFs of fainter samples (with luminosity threshold below L*), which is similar in trend to the case of fitting the 2PCFs. The decomposition of the model 3PCFs into contributions from different types of galaxy triplets shows that on small scales the dependence of the 3PCFs on triangle shape is driven by nonlinear redshift-space distortion (and not by the intrinsic halo shape) while on large scales it reflects the filamentary structure. The decomposition also reveals more detailed differences in the three models, which are related to the radial distribution, the mean occupation function, and the velocity distribution of satellite galaxies inside halos. The results suggest that galaxy 3PCFs can further help constrain the above galaxy–halo relation and test theoretical models.