A variety of samples made via different routes were investigated. Samples are nanostructured (average grain sizes are about 20 nm). The advantage of high-pressure (HP)-manufactured (2 GPa, 800-1050 degrees C, 1 h) MgB2 bulk is the possibility to get almost theoretically dense (1-2% porosity) material with very high critical current densities reaching at 20 K, in 0-1 T j(c) = 1.2 - 1.0 . 10(6) A/cm(2) (with 10% SiC doping) and j(c) = 9.2 - 7.3 10(5) A/cm(2) (without doping). Mechanical properties are also very high: fracture toughness up to 4.4 +/- 0.04 MPa . m(0.5) and 7.6 +/- 2.0 MPa . m(0.5) at 148.8 N load for MgB2 undoped and doped with 10% Ta, respectively. The HP-synthesized material at moderate temperature (2 GPa, 600 degrees C, 1 h) from B with high amount of impurity C (3.15%) and H (0.87%) has j(c) = 10(3) A/cm(2) in 8 T field at 20 K, highest irreversibility fields (at 18.4 K H-irr = 15 T) and upper critical fields (at 22 K H-C2 = 15 T) but 17% porosity. HP materials with stoichiometry near MgB12 can have T-c = 37 K and j(c) = 6 . 10(4) A/cm(2) at 0 T and H-irr = 5 T at 20 K. The spark plasma synthesized (SPS) material (50 MPa, 600-1050 degrees C 1.3 h, without additions), demonstrated at 20 K, in 0-1 T j(c) = 4.5 - 4 10(5) A/cm(2). Dispersed inclusions of higher magnesium borides, which are usually present in MgB2 structure and obviously create new pinning centers can be revealed by Raman spectroscopy (for the first time a spectrum of MgB7 was obtained). Tests of quench behavior, losses on MgB2 rings and material thermal conductivity show promising properties for fault current limiters. Due to high critical fields, the material can be used for magnets.