World models connect perception and decision-making in embodied intelligence by maintaining hidden state, anticipating consequences, comparing interventions, and adapting when execution departs from expectations. Although progress is often measured by visual fidelity, their value lies in improving behavior. Before reaching for a cup, a person anticipates its weight and resistance to grasping, shaping the hand before contact. Such anticipation is coarse and rarely pictorial, yet it guides action. This raises a central question: which predictive capabilities improve behavior? Existing surveys, organized by architecture, output modality, or application domain, leave this question implicit. We introduce three progressively stronger capability levels: Plausible models preserve task-relevant temporal, geometric, or physical structure; Controllable models additionally predict how interventions alter that structure; and Actionable models translate predictions into measurable gains in planning, action, learning, evaluation, verification, recovery, or data selection. We complement this hierarchy with a 3 x 4 matrix crossing geometry, physics, and action grounding with improvement loops centered on data, rewards, policies, and the model itself. Using this framework, we survey manipulation, navigation, locomotion, autonomous driving, and general embodied learning, tracing technical progressions, clarifying capability requirements, and examining datasets, benchmarks, and evaluation protocols. We identify challenges in long-horizon consistency, uncertainty calibration, causal intervention testing, latency, verification and recovery, and cross-embodiment transfer. This perspective shifts evaluation from visual plausibility toward whether predictions capture task-relevant state, reflect intervention effects, and improve the closed-loop behavior of embodied agents.

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