The knowledge about discrete quantum breathers, accumulated during the last two decades, is reviewed. "Prehistory" of the problem is described and some important properties differentiating localized and extended vibrational modes are outlined. The state of art of our understanding of the principal features of the quantum discrete breathers is presented. PACS numbers: The idea of a localization of vibrational energy due to anharmonic effects was put forward independently several times, in

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... nt fields, under different names, aiming at different applications. One of the most popular name for such a localized state during the last decade is breather. Quantization of such localized excitations has been also under study for quite a long time. This paper reviews the field with a special emphasis on the results related to the quantum breathers. Among the problems touched upon are quantization of breathers, separation between the quantum breathers, corresponding to classically localized states, and phonons -extended states, dynamical properties of breathers. The amount of the results related to quantum breathers is not large and the best has been done to make a possibly complete coverage of the field.