During the last decade, new unusual physical phenomena have been discovered in studying the optics of dielectric mesoscale particles of an arbitrary three-dimensional shape with the Mie size parameter near 10 (q ~ 10). The paper provides a brief overview of these phenomena from optics to terahertz, plasmonic and acoustic ranges. The different particle configurations (isolated, regular or Janus) are discussed, and the possible applications of such mesoscale structures are briefly reviewed herein in relation to the field enhancement, nanoparticle manipulation and super-resolution imaging. The number of interesting applications indicates to a new promising scientific direction emerged in optics, terahertz and acoustic ranges, and plasmonics. In this paper we present the authors' view of these problems.