This position paper proposes that the study of embodied cognitive agents, such as humanoid robots, can advance our understanding of the cognitive development of complex sensorimotor, linguistic, and social learning skills. This in turn will benefit the design of cognitive robots capable of learning to handle and manipulate objects and tools autonomously, to cooperate and communicate with other robots and humans, and to adapt their abilities to changing internal, environmental, and social conditions. Four key areas of research challenges are discussed, specifically for the issues related to the understanding of: 1) how agents learn and represent compositional actions; 2) how agents learn and represent compositional lexica; 3) the dynamics of social interaction and learning; and 4) how compositional action and language representations are integrated to bootstrap the cognitive system. The review of specific issues and progress in these areas is then translated into a practical roadmap based on a series of milestones. These milestones provide a possible set of cognitive robotics goals and test scenarios, thus acting as a research roadmap for future work on cognitive developmental robotics.