The U.S. Dept. of Energy's goal for multifilamentary (MF) strand intended for accelerator applications includes current densities, Jc, of up to 1200 A/mm2 in fields of 12-16 T. Although MgB2 films possess Bc2s at 0 K as high as 50 T, and Jcs (self field) as high as 107 A/cm2 have been reported, it may be several years before the above goal is attained in round MF MgB2 wire. But when its electrical properties become suitable MgB2's critical temperature of 39 K will be an added advantage in the accelerator magnets' high radiation environment. In the meantime, some near-term applications have been identified. Continually improving properties may enable MgB2 to be used in the windings of: (i) undulator magnet upgrades and replacements, (ii) wiggler magnets, (iii) light source bending magnet replacements, or (iv) some of the lower field solenoids of a proposed muon collider. Active research-and-development programs are yielding MgB2 strands with some remarkable properties: critical fields Bc2 and Birr in the ranges 28-33 T and 24-29 T, respectively, and Jc (4.2 K)s in the range 1.1-5.5times105 A/cm2, 5-2 T, respectively. Recent superconducting undulator (SCU) magnets are calling for Jc, BW (winding-field), B0 (beam-line field) combinations such as 1.1times105 A/cm2, 3.3 T, 1.0 T. These specifications, in the light of its existing performance levels, indicate MgB's potential suitability for the windings of SCU's. A model MgB2 coil-set is described