The reflectarray is a familiar antenna structure. It often allows replacing a thick, expensive reflector with a thin, flat, relatively low-cost device. A phase-shifting surface is a structure that is analogous to a reflectarray but utilizing transmission, replacing a lens antenna with a thin, flat, low-cost multilayer printed structure. The feature article by Nicolas Gagnon, Aldo Petosa, and Derek McNamara traces the history of phase-shifting surfaces, and describes the latest advances in this field. The article begins with an extensive historical review. This traces the development of the phase-shifting surface from metallic-waveguide and artificial-dielectric lenses, through microwave holography, to transmitarrays. The most recent designs have employed thin, multilayer designs with sub-wavelength elements. The authors then describe the design and operation of these phase-shifting surfaces in detail. A typical device consists of multiple layers of printed conductive elements. These elements are often strips or squares. The layers are spaced closely together, to make use of the mutual coupling between elements to expand the available range of phase shifts. The dimensions of the elements are varied along both planar axes to produce the desired phase shifts. The method of modeling the structure and the electric-field distribution in the structure are described. An equivalent-circuit model is developed. The results of experiments with several prototypes of free-standing lensing devices using phase-shifting surfaces are then reported. These demonstrated improved gain, bandwidth, and other performance features when compared to Fresnel-zone-plate antennas. The article concludes with a discussion of the major factors affecting future development of phase-shifting surfaces, including bandwidth, pattern discontinuities, the ability to scale designs with frequency, the effect of angle of incidence on operation, and practical factors affecting implementation. This article provides both an excellent overview of a fascinating antenna technology, and a very nice introduction to the current capabilities and future possibilities of that technology.