The paper considers all types of five-links geared linkages and presents a general synthesis method to design a desired transmission function. This function is usually very high non-linear. The method requires the computation of the centrodes of non-circular gear pairs with non-constant input and output motion and their corresponding transmission function. The different linkage types demand the investigation of relevant boundary conditions. A simple, general synthesis method is shown to solve the considered class of problems. The application improves the tuning of the non-linearity of a whole mechanism and the non-linear transmission ratio of the centrodes. The kinematical characteristics (transmission functions) of geared linkages can be also obtained. Numerical examples of typical mechanical problems are presented.