This paper reviews the potential of the newly discovered and unexpected 39 K superconductor MgB 2 as a material for accelerator magnet strands. Unlike the high temperature cuprate superconductors, intergranular current flow in MgB 2 is not obstructed by weak links, removing one serious obstacle to the fabrication of wires. The critical current density J c exceeds 10 5 A/cm 2 in 1 T field at 4.2 K and in self-field at 20 K. However, the compound is anisotropic and the perpendicular upper critical field H c2^ in present bulk samples is comparable to or less than that in Nb47wt.%Ti at 4.2 K. Thin films have shown significantly higher J c and H c2 . Magnetization J c values at 4.2 K exceed 10 6 A/cm 2 in many reports, and approach 10 7 A/cm 2 for films with T c close to 39 K. In alloyed films with small grain size and high resistivity, the irreversibility field H* (~0.85H c2 ) is 16 T at 4.2 K in perpendicular field (and double that in parallel field), and J c exceeds 10 5 A/cm 2 up to 10 T at 4.2 K and 5 T at 20 K. Simple wires, which achieve the properties of sintered bulk samples, are being made by the powder-in-tube route, indicating potential application capability, particularly if alloyed MgB 2 wires replicating thin film properties can be made.