Coloring in nature mostly comes from the inherent colors of materials, but it sometimes has a purely physical origin, such as diffraction or interference of light. The latter, called structural color or iridescence, has long been a problem of scientific interest. Recently, structural colors have attracted great interest because their applications have been rapidly progressing in many fields related to vision, such as the paint, automobile, cosmetics, and textile industries. As the research progresses, however, it has become clear that these colors are due to the presence of surprisingly minute microstructures, which are hardly attainable even by ultramodern nanotechnology. Fundamentally, most of the structural colors originate from basic optical processes represented by thin-film interference, multilayer interference, a diffraction grating effect, photonic crystals, light scattering, and so on. However, to enhance the perception of the eyes, natural creatures have produced various designs, in the course of evolution, to fulfill simultaneously high reflectivity in a specific wavelength range and the generation of diffusive light in a wide angular range. At a glance, these two characteristics seem to contradict each other in the usual optical sense, but these seemingly conflicting requirements are realized by combining appropriate amounts of regularity and irregularity of the structure. In this Review, we first explain the fundamental optical properties underlying the structural colors, and then survey these mysteries of nature from the viewpoint of regularity and irregularity of the structure. Finally, we propose a general principle of structural colors based on structural hierarchy and show their up-to-date applications.