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We present the theory of charge transport in contacts with multiband unconventional and topological superconductors. This theory based on recently proposed microscopic tight-binding approach [1,2]. This approach [1,2] takes into account the complex excitation spectrum of these superconductors, their multiband Fermi surface, the anisotropy of the superconducting order parameter as well as interband and intervalley scattering at the boundaries. We have considered theoretically break junctions of multiband Fe-based superconductor (FeBS) and contacts between usual single-band s-wave superconductor and FeBS (S-Sp junctions). We also calculated current-voltage characteristics of contacts between doped superconducting insulators CuxBi2Se3 and normal metal, based on microscopic approach [1,2]. We have demonstrated that investigation of current-voltage characteristics of FeBS break junctions provides the possibility to distinguish possible s++ and s+- symmetries of the order parameter in FeBS only in the case of tunneling in a-b plane of FeBS. The investigation of c-oriented FeBS break junctions provide the information about magnitudes of gaps in FeBS and about the anisotropy of gaps. We also have calculated phase dependencies of the Josephson current and temperature dependencies of critical Josephson current across S-Sp junctions for different directions of current relative to the crystallographic axes of FeBS and different lengths of an insulator layer. We have demonstrated the relatively large contribution of the second harmonic to Josephson current-phase relation of a-b oriented S-Sp junctions. We have confirmed microscopically the recently proposed experimental scheme to determine the symmetry of the order parameter in FeBS [3] for c-oriented FeBS Josephson junctions. Based on our microscopic approach [1,2] we have calculated current-voltage characteristics of contacts between doped superconducting insulators CuxBi2Se3 and normal metal and compare our results with previous phenomenological calculations, presented in [4].