The cooperation between humans and robots is becoming increasingly important in our society. Consequently, there is a growing interest in the development of models that can enhance the interaction between humans and robots. A key challenge in the Human-Robot Interaction (HRI) field is to provide robots with cognitive and affective capabilities, developing architectures that let them establish empathetic relationships with users. Several models have been proposed in recent years to solve this open-challenge. This work provides a survey of the most relevant attempts/works. In details, it offers an overview of the architectures present in literature focusing on three specific aspects of HRI: the development of adaptive behavioural models, the design of cognitive architectures, and the ability to establish empathy with the user. The research was conducted within two databases: Scopus and Web of Science. Accurate exclusion criteria were applied to screen the 1007 articles found (at the end 30 articles were selected). For each work, an evaluation of the model is made. Pros and cons of each work are detailed by analysing the aspects that can be improved so that an enjoyable interaction between robots and users can be established.