Ion beams in the kiloelectronvolt energy range impinging on crystal surfaces are atomic-size probes of the nanoscale magnifying and focusing "lenses" formed by the periodic structures, These lenses disperse the scattered and recoiled atom trajectories into macroscopic projections according to their velocities as a function of projectile/target atom masses and deflection angles. The resulting patterns reflect the near-surface interatomic vectors. The spatially and temporally resolved images of these patterns can now be acquired by means of new developments in ion scattering spectrometry, providing direct, element-specific. real-space projections of surface structure. The resulting scattering and recoiling imaging spectrometry (SARIS) provides unique capabilities for analyzing surface composition and structure and for investigating the interactions of ions with surfaces. A classical ion trajectory program called scattering and recoiling imaging code (SARIC) that simulates the spatial and time intensity distributions of the SARIS images has been developed. A quantitative comparison of experimental and simulated images is achieved by means of a two-dimensional reliability (R) factor. Examples of applications include experimental stereographic projections of the Ni(110) and Pt(111) surfaces, the termination layer of the CdS(0001) surface, the chemisorption of Cl-2 on Ni(110) and H-2 On Pt(111), and the atomic lenses in the Pt(111) surface.