In scattering and radiation experiments in electromagnetics, optics and acoustics, the scattered and radiated fields are frequently measured in theradiation zoneorfar-field, many wavelengths away from the source These fields arehomogeneousorpropagatingwith a1/r1/rfall-off. But in thenear-fieldwithin a distance of the order of a wavelength from the source, the fields are quasi-static and are calledinhomogeneousandnon-propagating, decaying as1/r3.1/r^3.It is the phenomenon ofevanescenceand the spatial spectrum of the near-field is dominated byevanescent waves. These waves are unobservable unlessfrustrated, i.e., thwarted by some object. They are and are not waves at the same time and have characteristics that seem to defy common sense. They can skim along a flat surface as regular propagating waves while exponentially decaying in the perpendicular direction. Yet, they are known to be present at infinity along some special directions. Whether their presence away from the surface is location or source dependent, is unclear. Moreover, how the propagating modes disentangle themselves from the quasi-static near-field and arrive at the radiation zone is alsosub judice. Controversies around these questions abound. Having a purely imaginary wavenumber, the evanescent waves were once thought to be of mathematical interest only, with no particular physical imports. But in the physics of near-field electromagnetics, optics, photonics and acoustics, with a number of breakthrough technological applications, evanescent waves occupy central position. Owing to their close proximity to the source, they are carriers of its high frequency information and are instrumental in achieving resolutions far exceeding the diffraction-limited classical limit. Evanescent modes are, therefore, crucial in high resolution imaging and inverse reconstructions of unknown objects. They are also highly relevant to scattering involving different media, high resolution spectroscopy and even high density data storage. In quantum mechanics, the evanescent waves are thought to be virtual photons and the phenomenon of evanescence as quantum tunneling. It is also known that in non-relativistic physics, the evanescent modes obey the macroscopic principle of causality of cause preceding effect. Evanescence has been discussed in a number of monographs and journal articles. The primary focus is overwhelmingly on optics. Our objective in this Chapter is to give an overall account of evanescence including simple, physical, pictures of how it can arise in unrelated practical situations.