The rich diversity and the fundamental character of the essential theo
retical problems associated with it have given band theory a width of
interest which contrasts strongly with the apparent narrowness of its
subject matter. This review, dealing mainly with the classical band st
ructures of periodic elastic and acoustic binary systems, offers brief
ly a systematic survey of the historical development of the principles
, tools, and applications of band theory for electrons, phonons, photo
ns, and vibrations giving what may be called the ''background'' to the
more recent developments in the fields of photonic and phononic band-
gap crystals. Attention is given to survey the physical conditions req
uired to achieve the complete spectral gaps within which the respectiv
e propagating modes are utterly forbidden irrespective of the directio
n of propagation. The existence of complete spectral gaps for cleverly
synthesized photonic crystals guarantees the observability of classic
al Anderson localization of photons and the influence on the spontaneo
us emission which was, until the 1980's, often regarded as a natural a
nd uncontrollable phenomenon. The phononic band-gap crystals, on the o
ther hand, offer the feasibility of constructing the ultrasound filter
s, polarization filters, and improvements in designing the transducers
, as well as the observability of classical elastic or acoustic wave l
ocalization. Abiding by the central theme of the review, numerous theo
retical results on the band structure related problems for periodic el
astic and acoustic binary systems have been gathered and reviewed. Thi
s survey is preceded by a detailed mathematical machinery that provide
s the reader with numerous useful analytical results applicable to a w
ide range of systems of varying interest. Finally, the report conclude
s with a summary of anticipated implications of photonic and phononic
band-gap crystals and proposes some interesting relevant problems conc
erned with the spectral gaps and the classical wave localization. Our
satisfaction in writing this review, like any other review which cover
s a considerably longer period, was to reach a reasonably self-contain
ed unity by wanting to ''leave nothing unexplained''. The background p
rovided is believed to make less formidable the task of future writers
of reviews in this rather general field and hence enable them to deal
more readily with particular aspects of the subject, or with recent a
dvances in those directions in which notable progress may have been ma
de.