More than 30-40 years ago there was established that spin-orbit interaction (SOI) plays an important role in the formatting of band structure of solid state matter. Among different qualitative features induced by SOI the band inversion of electronic spectrum near the Fermi level has been discovered. Such type of electronic spectrum was identified in different type of semimetalic and narrow-gap semiconductors Bi1-xSbx, Pb1xSnxTe, Bi2Te3, HgTe, TlBiTe2 etc. In the context of low dimensional structure investigations the band spectrum inversion was shown to generate new type of interface gapless states with linear spectrum at the heterocontact boundaries. Last years investigations have reopened the interest to materials with inverted spectra. Due to new type of the symmetry break like that characteristic for the integer and fractional quantum Hall effects the electronic states was shown to have topological nature and materials have been named toplogical insulators (TI). Thus in TI a new state of matter appear, distinguished from a regular band insulator by a nontrivial time-reversal topological invariant, which characterizes its bandstructure. In results new physics and phenomena related to this states have greatly emerged. Several of such new TI properties are reviewed in the first part of the paper as well as some old observed properties of materials with band inversion. The most robust observable consequence of a nontrivial topological character of these materials is the presence of gapless helical edge states (interface states of inverted heterocontacts), whose gapless states is protected by time-reversal symmetry and is thus robust to perturbations that do not break this symmetry. Like the Hall state the “bulk” of the electron gas of TI is an insulator, but along its suface, the states can be gapless. Within a certain parameter range the surface states of TI are well described by a Dirac cone, allowing for parallels with graphene and relativistic physics, and prohibiting backscattering. At the same time the spectrum and characteristics of topological surface states (TSS) depending on geometrical configuration can be manipulates by different factors: electrical and magnetic fields, strain and deformation ets. For this reason TI are being explored with a view towards applications, as a potential platform for tailoring nanostructures and nanomaterials properties.This topics cover the second part of the paper. The evolution of the topological states in dependence of layer thickness and others factors are highlighted for quantun well and superlattice structures based on Bi1-xSbx, Pb1-xSnxTe, Bi2Te3 in the framework of two-band effective mass method. In the superlattice structures like PbTe/SnTe with layer thickness a and b respectively the state of the topological insulator can be realized. The phase diagram of the band and topological insulator states are established in dependence of the semiconductor gaps and thickness. The gapless electronic states with Dirac like linear spectrum were revealed to occur when EgPbTe*a= EgSbTe* b. Such superlattice structures can be considered as a new type of metamaterial of semiconductor layers and metallic sheets. The plasmonic and metamaterial characteristics of such layered structures are discussed. Flat-lens focusing of electrons on the surface of a topological insulator Bi2Te3 is analysed. The early studied interface states in inverted heterocontact with magnetic ordering are reanalyzed in the context of recently discovered antiferromagnetic TI. The occurrence of interface ferromagnetism is demonstrated. The last part of the paper deals with behavior of TSS if TI nanostructures in the nanowire and nanotube like configuration. The TSS of cylindrical nanowires and topological insulator Bi2Te3 with cylindrical pores are studied. The developed recently low-energy approach for bulk Bi2Te3 is used to highlight TSS on the cylindrical surface. The TSS forming inside the bulk gap corresponds to one dimensional bands indexed by total angular momentum. For nanowire or nanopore of radius R, the wavefunction to vanish at the boundary r = R is required, which is automatically ensured by expanding in the orthonormal set of radial Bessel functions Jm or Ym with integer m. In comparison with gapless character os TSS of flat surface all TSS modes of cylindrical surface have a finite gap described qualitatively by relations Egs ~v/R. In results nanowire and nanopore composites of TI have distinct from layered ones pecularities and several are discussed in the paper. Some aspects of transport through TSS are discussed: anomalous Aharonov-Bohm conductance oscillations; magnetic quantum oscillations, edge accumulation and currents of moment. Thermoelectric aspect of TSS are discussed in the context of TI materials Bi2Se3 and Bi2Te3 knowing as the best thermoelectrics.