We report novel results on studies of giant magnetoimpedance (GMI) effect and magnetic properties in ultrathin amorphous glass-coated microwires with nearly zero and negative magnetostriction constant and metallic nucleus diameter ranging within 6 and 12 mu m. Off-diagonal MI curve of Co_(67.1)Fe_(3.8)Ni_(14.5)Si_(14.5)B_(11.5)Mo_(1.7) microwire exhibits asymmetrical shape with almost linear growth within the field range from -H_m to H_m associated with the anisotropy field. Co_(74)B_(13)Si_(11)C_2 microwire possesses hysteretic off-diagonal MI curve. Current annealing significantly changes off-diagonal MI curves, especially in the case of Co_(74)B_(13)Si_(11)C_2 microwire.
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