The torsion effect on the field and current driven magnetization reversal and the associated domain wall velocity in cylindrical amorphous and nanocrystalline glass-coated microwires is reported. Samples from three representative compositions have been investigated: (1) amorphous Fe77.5Si7.5B15 with positive magnetostriction, λ ≅ 25 × 10−6, (2) amorphous Co68.18Fe4.32Si12.5B15 with nearly zero negative magnetostriction, λ ≅ −1 × 10−7, and (3) nanocrystalline Fe73.5Si13.5B9Cu1Nb3 (FINEMET) with small positive magnetostriction, λ ≅ 2.1 × 10−6, all having the diameter of the metallic nucleus, d, of 20 µm and the glass coating thickness, tg, of 11 µm. The results are explained through a phenomenological interpretation of the effects of applied torque on the anisotropy axes within the microwires with different characteristics. Among all the complex mechanical deformations caused by the application of torque on magnetic microwire samples, the most important are the axial compression – for axial field-driven domain wall motion, and the circumferential tension – for electrical current/circumferential field-driven domain wall motion. The Co68.18Fe4.32Si12.5B15 microwire, annealed at 300 °C for 1 hour and twisted at 168 Rad/m exhibits the optimum characteristics, e.g. the lowest switching current (down to 9 mA~2.9 × 10−3 A/cm2) and the largest domain wall velocity (up to 2300 m/s).