Reach-scale at-a-station hydraulic geometry (AHG) relationships are power laws that describe variations of reach-averaged water depth, wetted width, and current velocity in stream reaches when discharge varies. Modeling AHG exponents is important, because the variations of hydraulics with discharge in stream networks influence physical habitats of aquatic species, biodiversity, water temperature, nutrient fluxes, and sediment transport. Theoretical approaches indicated that AHG exponents should depend on topographic descriptors of cross sections and roughness elements. Empirical approaches suggested that AHG exponents partly depend on hydraulic characteristics measured at a single discharge. We used a unique data set of AHG observed in 812 stream reaches (obtained from measurements at several discharge rates or from hydrodynamic models) to (1) test the consistency of theoretical and empirical predictions of AHG exponents and (2) test the generality of AHG predictions across rivers of different countries with variable landscape settings. We found that observed AHG depended on topographic predictors (describing cross-section shape and substrate size) as expected from theory. However, AHG exponents were better predicted by empirical hydraulic characteristics of reaches: the ratio of wetted width to bankfull width and the reach Froude number. The effects of hydraulic variables were consistent with those of topographic predictors. In addition, relations between AHG and hydraulic predictors were significant and with similar direction in different data sets. Despite limited explanatory power, our results help identifying general drivers of AHG exponents. Their application still requires measurements at a single discharge rate but open perspectives of generalized AHG prediction by remote sensing.