A study of magnetic hysteresis and giant magnetoimpedance (GMI) in amorphous glass-covered Co–Si–B and Co–Mn–Si–B wires is presented. The wires, about 10 μm in diameter, were obtained by a glass-coated melt spinning technique. Samples with positive magnetostriction (MS) have a rectangular bistable hysteresis loop. A smooth hysteresis loop is observed for wires with nearly zero MS. When the MS is negative, almost no hysteresis is observed. The GMI was measured in the frequency range between 20 Hz and 30 MHz. The shapes of the impedance versus field curves are qualitatively similar to each other for both positive and zero MS samples. The impedance is maximum at zero field, and decreases sharply in the range of 10–20 Oe. For the negative MS wires, when the driving current is small, the impedance is maximum at a finite external field. The position of the maximum approaches zero with increasing current. The contributions of the moment rotation and domain wall motion in the three cases are discussed.