In solid‐state physics, the phenomenon of localization is usually perceived as arising fromextrinsicdisorder that breaks the discrete translational invariance of the perfect crystal lattice. Familiar examples include the localized vibrational phonon modes around impurities or defects (such as atomic vacancies or interstitial atoms) in crystals and Anderson localization of electrons in disordered media. The usual perception among solid‐state researchers is that, in perfect lattices—those free of extrinsic defects—phonons and electrons exist only in extended, plane wave states. That notion extends to any periodic structure, such as a photonic crystal or a periodic array of optical waveguides. Such firmly entrenched perceptions were severely jolted in the late 1980s by the discovery thatintrinsiclocalized modes (ILMs), also known as discrete breathers (DBs), are, in fact, typical excitations in perfectly periodic but strongly nonlinear systems.