Children are keen users of new technologies and new technologies can provide interesting opportunities to enrich children's experience, e.g., for educational and therapeutic purposes. As children are not small adults, it is necessary to research their specific needs and develop systems that address them. The Aliz-e project develops embodied cognitive robots for adaptive social interaction with young users over several sessions in real-world settings. We demonstrate a conversational system developed in the Aliz-e project using the Nao robot. It engages a user in the following activities: - quiz: the child and the robot ask each other series of multiple-choice quiz questions from various domains, the robot provides evaluation feedback; - imitation: either the child or the robot presents a sequence of simple arm poses that the other tries to memorize and imitate; - dance: the robot explores various dance moves with the child and then teaches the child a dance sequence according to its abilities. Besides activity-specific conversation, the interactions involve also a social component (greetings, introductions). During an activity, the robot provides performance feedback to the user. The social aspect here requires careful handling of the evaluation process so as not to discourage the user with negative feedback. As the system is designed to have multiple encounters with a user, the robot's behavior differs in various aspects from the first session (meeting for the first time) to the subsequent sessions ({"}knowing{"} the user and their performance).