By introducing a new spin-Bose transformation which incorporates the single-site spin-states mixing effect self-consistently, we establish a unified spin-wave approach, applicable to an arbitrary spin-number case and general spin configurations, for quantum spin systems with single-ion anisotropies. The conventional Holstein-Primakoff (HP) method completely fails for such systems with easy-plane anisotropy and sometimes fails for those with easy-axis anisotropy, while these difficulties have been overcome successfully by the new method. In some limiting cases, the present method recovers the results of old theories which are valid in such cases. Applications to magnetic multilayers show that the new method is useful to remove the unphysical instability predicted by the conventional HP method.