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The electromagnetic waves interaction with composite media attract great attention of researches for many decades due to its relevance to problems in condensed matter physics, optics, photonics, plasmonics, and chemistry. During last two decades, metamaterials and photonic crystals have been in the top of research due to their unprecedented possibilities to manipulate the electromagnetic parameters of both materials and electromagnetic waves. The review is devoted to different types of artificial composite media, their classification, discussion of their unique characteristics and ways of control of their dispersion. Comprehensive review of the electromagnetic properties of periodic and aperiodic planar Bragg reflectors (that is, photonic crystals) and planar Bragg reflective waveguides is carried out. The dispersion features of Bragg reflective waveguides with both periodic and aperiodic arrangements of layers in their claddings are discussed and methods of their control are presented. It was found that an aperiodic configuration of cladding of Bragg reflection waveguide could give rise to exceptionally strong mode selection and tuning the polarization-discrimination effects. On the other hand, artificial media alled metamaterials (and especially, hyperbolic metamaterials) created using subwavelength resonant building blocks, are also useful for both controlling light propagation and dispersion management. They can be easily made by alternating dielectric and metal layers or by embedding arrays of parallel metallic rods in a dielectric matrix. This review discusses a particular example of hyperbolic metamaterial, represente by a superlattice consisting of ferrite and semiconductor layers, which is influenced by an external static magnetic field. Within the framework of the effective medium theory, such an artificial structure can be reduced to a homogenized medium, which is described the effective permittivity and permeability tensors. Due to the components of both tensors show significant sensitivity to the external magnetic field, these artificial structures can exhibit the great variety of high-frequency properties. For instance, it is observed that in the case when specific conditions related to the superlattice`s constitutive parameters and filling factor are satisfied, the regions of existence of the bulk and surface polaritons can totally overlap. Besides, it is found out that in an extremely anisotropic medium, the dispersion characteristics of extraordinary bulk waves exhibit a number of unusual behaviors, including atypical topological transitions of isofrequency surfaces. The conditions for appearance of mono-hyperbolic, bi-hyperbolic, tri-hyperbolic and tetra-hyperbolic-like forms of isofrequency surfaces are also discussed.