The synthesis of MgB2-based materials under high pressure gave the possibility to suppress the evaporation of magnesium and to obtain near theoretically dense nanograined structures with high superconducting, thermal conducting, and mechanical characteristics: critical current densities of 1.8-1.0×106 A/cm2 in the self-field and 103 A/cm2 in a magnetic field of 8 T at 20 K, 5-3×105 A/cm2 in self-field at 30 K, the corresponding critical fields being Hc2=15 T at 22 K and irreversible fields Hirr=13 T at 20 K, and Hirr=3.5 T at 30 K, thermal conduction of 53±2 W/(m{dot operator}K), the Vickers hardness HV=10.12±0.2 GPa under a load of 148.8 N and the fracture toughness K1 C=7.6±2.0 MPa{dot operator}m0.5 under the same load, the Young modulus E=213 GPa. Estimation of quenching current and AC losses allowed the conclusion that high-pressure-prepared materials are promising for application in transformer-type fault current limiters working at 20-30 K.