The paper provides a brief outline of methods for automated synthesis of mechanical systems in general, with primary focus on application domains involving kinematics of mechanical motions. Two application domains described in this paper, mechanisms and machine tool design, employ methods of selecting and assembling appropriate building blocks to satisfy given functional (motion) specifications. This is called Synthesis by Composition. The functionality of the precompiled building blocks as well as the design task specifications are captured mathematically in a unified representation scheme. Generation of alternate designs is accomplished by manipulation of matrices representing functions. The paper also presents another method of Synthesis by Elimination to automatically generate topologies of flexible structures which satisfy given input/output forces and displacements. The methodology presented in this paper has been successfully applied to computational synthesis of mechanisms, machine tools, compliant mechanisms, and differentials. Various applications of these methods to automotive, aerospace, and medical and MEMS fields will be presented at the symposium.