Dexterous intelligence -- the ability to perform complex interactions with multi-fingered hands -- is a pinnacle of human physical intelligence and emergent higher-order cognitive skills. However, contrary to Moravec's paradox, dexterous intelligence in humans appears simple only superficially. Many million years were spent co-evolving the human brain and hands including rich tactile sensing. Achieving human-level dexterity with robotic hands has long been a fundamental goal in robotics and represents a critical milestone toward general embodied intelligence. In this pursuit, computational sensorimotor learning has made significant progress, enabling feats such as arbitrary in-hand object reorientation. However, we observe that achieving higher levels of dexterity requires overcoming very fundamental limitations of computational sensorimotor learning. I develop robot learning methods for highly dexterous multi-fingered manipulation by directly addressing these limitations at their root cause. Chiefly, through key studies, this disseration progressively builds an effective framework for reinforcement learning of dexterous multi-fingered manipulation skills. These methods adopt structured exploration, effectively overcoming the limitations of random exploration in reinforcement learning. The insights gained culminate in a highly effective reinforcement learning that incorporates sampling-based planning for direct exploration. Additionally, this thesis explores a new paradigm of using visuo-tactile human demonstrations for dexterity, introducing corresponding imitation learning techniques.