Accuracy is one of the most important key indices to evaluate multi-axis systems' (MAS's) characteristics and performances. The accuracy of MAS's such as machine tools, measuring machines and robots is adversely affected by various error sources, including geometric imperfections, thermal deformations, load effects, and dynamic disturbances. The increasing demand for higher dimensional accuracy in various industrial applications has created the need to develop cost-effective methods for enhancing the overall performance of these mechanisms. Improving the accuracy of a MAS by upgrading the physical structure would lead to an exponential increase in manufacturing costs without totally eliminating geometrical deviations and thermal deformations of MAS components. Hence, the idea of reducing MAS's error by a software-based alternative approach to provide real-time prediction and correction of geometric and thermally induced errors is considered a strategic step toward achieving the full potential of the MAS. This paper presents a structured approach designed to improve the accuracy of Cartesian MAS's through software error compensation. Four steps are required to develop and implement this approach: (i) measurement of error components using a multidimensional laser interferometer system, (ii) tridimensional volumetric error mapping using rigid body kinematics, (iii) volumetric error prediction via an artificial neural network model, and finally (iv) implementation of the on-line error compensation. An illustrative example using a bridge type coordinate measuring machine is presented.