This work focuses on generating multiple coordinated motor skills for intelligent systems and studies a Multi-Expert Synthesis (MES) approach to achieve versatile robotic skills for locomotion and manipulation. MES embeds and uses expert skills to solve new composite tasks, and is able to synthesise and coordinate different and multiple skills smoothly. We proposed essential and effective design guidelines for training successful MES policies in simulation, which were deployed on both floating- and fixed-base robots. We formulated new algorithms to systematically determine task-relevant state variables for each individual experts which improved robustness and learning efficiency, and an explicit enforcement objective to diversify skills among different experts. The capabilities of MES policies were validated in both simulation and real experiments for locomotion and bi-manual manipulation. We demonstrated that the MES policies achieved robust locomotion on the quadruped ANYmal by fusing the gait recovery and trotting skills. For object manipulation, the MES policies learned to first reconfigure an object in an ungraspable pose and then grasp it through cooperative dual-arm manipulation.