This dissertation considers Open-world Robot Manipulation, a manipulation problem where a robot must generalize or quickly adapt to new objects, scenes, or tasks for which it has not been pre-programmed or pre-trained. This dissertation tackles the problem using a methodology of efficient sensorimotor learning. The key to enabling efficient sensorimotor learning lies in leveraging regular patterns that exist in limited amounts of demonstration data. These patterns, referred to as ``regularity,'' enable the data-efficient learning of generalizable manipulation skills. This dissertation offers a new perspective on formulating manipulation problems through the lens of regularity. Building upon this notion, we introduce three major contributions. First, we introduce methods that endow robots with object-centric priors, allowing them to learn generalizable, closed-loop sensorimotor policies from a small number of teleoperation demonstrations. Second, we introduce methods that constitute robots' spatial understanding, unlocking their ability to imitate manipulation skills from in-the-wild video observations. Last but not least, we introduce methods that enable robots to identify reusable skills from their past experiences, resulting in systems that can continually imitate multiple tasks in a sequential manner. Altogether, the contributions of this dissertation help lay the groundwork for building general-purpose personal robots that can quickly adapt to new situations or tasks with low-cost data collection and interact easily with humans. By enabling robots to learn and generalize from limited data, this dissertation takes a step toward realizing the vision of intelligent robotic assistants that can be seamlessly integrated into everyday scenarios.