The authors provide an extensive review of this field, outlining its multidisciplinary history, presenting a state-of-the-art review of current methods and applications, a description of the phenomena observed and projections for future research directions. This note provides additional comments and interpretation in various areas. Wave motion in periodic structures is discussed, with the emphasis being on multicoupled and continuous systems. The natural frequencies of finite structures are considered, including higher frequency issues such as asymptotic modal density and how periodic subsystems can be included in statistical energy analysis models. Comments are made on various computational and numerical issues. Media with periodic arrays of internal resonators—commonly referred to as acoustic metamaterials—are known to exhibit a stop band in the sub-Bragg frequency region around the resonator natural frequency: it is noted that the same effect can be produced by just a single resonator, rather than requiring a periodic array. Phenomena occasionally referred to in the literature as involving “negative group velocity” or “negative mass” are discussed and alternative physical interpretations provided.