The relevant pinning centers of Abrikosov vortices in MgB2–based materials are oxygen-enriched Mg-B-O inclusions or nanolayers and inclusions of MgBx (x>4) phases. The high critical current densities, jc, of 106 and 103A/cm2 at 1 and 8.5 T, respectively, at 20 K canbe achieved in polycrystalline materials (prepared at 2 GPa) containing a large amount of admixed oxygen. Besides, oxygen can be incorporated into the MgB2 structure in small amounts (MgB1.5O0.5), which is supported by Auger studies and calculations of the DOS and the binding energy. The jc of melt textured YBa2Cu3O7-(or Y123)-based superconductors [MT-YBaCuO) depends not only on the perfectness of texture and the amount of oxygen in theY123 structure, but also on the density of twins and micro-cracks formed during the oxygenation (due to shrinking of the c-lattice parameter). The density of twins and microcracks increases with the reduction of the distance between Y2BaCuO5 (Y211) inclusions in Y123. At 77 K jc=8⋅104 A/cm2 in self-field and jc=103 A/cm2 at 10 Т were found in materials oxygenated at 16 MPa for 3 days with a density of twins of 22–35 per μm (thickness of thelamellae: 45-30 nm) and a density of micro-cracks of 200–280 per mm. Pinning can occur atthe points of intersection between the Y123 twin planes and the Y211 inclusions. MT-YBaCuOat 77 K can trap 1.4 T (38×38×17 mm, oxygenated at 0.1 MPa for 20 days) and 0.8 T(16 mm in diameter and 10 mm thick with 0.45 mm holes oxygenated at 10 MPa for 53 h). Thesensitivity of MgB2 to magnetic field variations (flux jumps) complicates estimates of the trapped field. At 20 K 1.8 T was found for a block of 30 mm in diameter and a thickness of 7.5mm and 1.5 T (if the magnetic field was increased at a rate of 0.1 T) for a ring with dimensions 24×18 mm and a thickness of 8 mm.