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With the increasing demand for the dexterity of robotic operation, dexterous manipulation of multi-fingered robotic hands with reinforcement learning is an interesting subject in the field of robotics research. Our purpose is to present a comprehensive review of the techniques for dexterous manipulation with multi-fingered robotic hands, such as the model-based approach without learning in early years, and the latest research and methodologies focused on the method based on reinforcement learning and its variations. This work attempts to summarize the evolution and the state of the art in this field and provide a summary of the current challenges and future directions in a way that allows future researchers to understand this field.