This study summarizes advances in radar backscattering from refractive index irregularities in the clear air and its relationship to atmospheric structure and turbulence. This recent discipline of radar meteorology originated from radar angel studies and the efforts to explain these mysterious echoes. The significance of specular atmospheric radar reflections now appears limited, and consequently this study is devoted entirely to radar backscattering from turbulent fluctuations in the clear-air refractive index. Two regimes of atmospheric turbulence are distinguished: (1) in the convective domain, backscattering from water vapor fluctuations outlines regular, three-dimensional clear-air structures associated with buoyant, moist air parcels; (2) in the stable regime, horizontally stratified clear-air echoes originate from water vapor and/or temperature perturbations due to wind-shear-generated turbulent mixing in zones of enhanced static stability. Theoretical relationships are given between the radar reflectivity and the refractive index microstructure for the general case, for the isotropic case, and for the inertial subrange. The connection between the microstructure and the atmospheric mean fields is illustrated for a simplified situation in the stable regime; this relationship provides insight to the radar backscattering from clear-air turbulence (CAT). It is concluded that strong CAT will be associated with zones of increased refractive index variability and enhanced radar returns. A brief résumé of experimental work illustrates the application of radar methods in clear-air research and summarizes characteristic features of clear-air structures.