Social robots, owing to their embodied physical presence in human spaces and the ability to directly interact with the users and their environment, have a great potential to support children in various activities in education, healthcare and daily life. Child-Robot Interaction (CRI), as any domain involving children, inevitably faces the major challenge of designing generalized strategies to work with unique, turbulent and very diverse individuals. Addressing this challenging endeavor requires to combine the standpoint of the robot-centered perspective, i.e. what robots technically can and are best positioned to do, with that of the child-centered perspective, i.e. what children may gain from the robot and how the robot should act to best support them in reaching the goals of the interaction. This article aims to help researchers bridge the two perspectives and proposes to address the development of CRI scenarios with insights from child psychology and child development theories. To that end, we review the outcomes of the CRI studies, outline common trends and challenges, and identify two key factors from child psychology that impact child-robot interactions, especially in a long-term perspective: developmental stage and individual characteristics. For both of them we discuss prospective experiment designs which support building naturally engaging and sustainable interactions.