Creating autonomous robots that can actively explore the environment, acquire knowledge and learn skills continuously is the ultimate achievement envisioned in cognitive and developmental robotics. Their learning processes should be based on interactions with their physical and social world in the manner of human learning and cognitive development. Based on this context, in this paper, we focus on the two concepts of world models and predictive coding. Recently, world models have attracted renewed attention as a topic of considerable interest in artificial intelligence. Cognitive systems learn world models to better predict future sensory observations and optimize their policies, i.e., controllers. Alternatively, in neuroscience, predictive coding proposes that the brain continuously predicts its inputs and adapts to model its own dynamics and control behavior in its environment. Both ideas may be considered as underpinning the cognitive development of robots and humans capable of continual or lifelong learning. Although many studies have been conducted on predictive coding in cognitive robotics and neurorobotics, the relationship between world model-based approaches in AI and predictive coding in robotics has rarely been discussed. Therefore, in this paper, we clarify the definitions, relationships, and status of current research on these topics, as well as missing pieces of world models and predictive coding in conjunction with crucially related concepts such as the free-energy principle and active inference in the context of cognitive and developmental robotics. Furthermore, we outline the frontiers and challenges involved in world models and predictive coding toward the further integration of AI and robotics, as well as the creation of robots with real cognitive and developmental capabilities in the future.