To avoid complicated force control system in precision manipulation, compliant constant force/ torque mechanisms are proposed and developed continuously since the 1990s. The inherent nonlinearities due to varying stiffness raise high requirements for mechanism design, kinetostatic modeling and structural optimization. This makes the synthesis of compliant constant force/ torque mechanism a multidisciplinary and multiplex task. In this work, a comprehensive survey on synthesis methods of the constant force mechanisms is conducted. Three categories of synthesis methods are summarized, i.e., the Rigid-body Replacement approach, the Building Block approach and the Structural Optimization approach. For each category, the corresponding modeling and optimization methods are introduced in details. The presented survey provides overall perspectives on current status and future challenges in this field for designers and researchers.