Achieving human-like dexterous robotic manipulation remains a central goal and a pivotal challenge in robotics. The development of Artificial Intelligence (AI) has allowed rapid progress in robotic manipulation. This survey summarizes the evolution of robotic manipulation from mechanical programming to embodied intelligence, alongside the transition from simple grippers to multi-fingered dexterous hands, outlining key characteristics and main challenges. Focusing on the current stage of embodied dexterous manipulation, we highlight recent advances in two critical areas: dexterous manipulation data collection (via simulation, human demonstrations, and teleoperation) and skill-learning frameworks (imitation and reinforcement learning). Then, based on the overview of the existing data collection paradigm and learning framework, three key challenges restricting the development of dexterous robotic manipulation are summarized and discussed.