MgB2-based nanostructural materials with rather high oxygen concentration (5-14 wt.%) and dispersed grains of higher borides (MgB12, MgB7) high-pressure (2 GPa or 30 MPa) synthesized (in-situ) or sintered (ex-situ) demonstrated high superconducting characteristics (critical current density, jc, up to 1.8-1.0106 A/cm2 in the self magnetic field and 103 in 8 T field at 20 K, 3-1.5105 A/cm2 in the self field at 35 K, upper critical field up to HC2 = 15 T at 22 K, field of irreversibility Hirr =13 T at 20 K). The additives (Ti, SiC) and synthesis or sintering temperature can affect the segregation of oxygen and formation of oxygen-enriched Mg-B-O inclusions in the material structure, thus reducing the amount of oxygen in the material matrix as well as the formation of higher borides grains, which affects an increase of the critical current density. The record high HC2 and Hirr have been registered for the material high-pressure (2 GPa) synthesized from Mg and B at 600 oC having 17% porosity and more than 7 wt.% of oxygen. The attained values of the critical current, AC losses and thermal conductivity make the materials promising for application for fault current limiters and electromotors. The structural and superconducting (SC) characteristics of the material with matrix close to MgB12 in stoichiometry has been studied and the SC transition Tc=37 K as well as jc= 5×104 A/cm2 at 20 K in the self field were registered, its Raman spectrum demonstrated metal-like behavior.