Social robots, also known as service or assistant robots, have been developed to improve the quality of human life in recent years. Socially assistive robots (SAR) are a special type of social robots that focus on providing support through social interaction. The design of socially capable and intelligent robots can vary, depending on the target user groups. In this work, I assess the effect of socially assistive robots' roles, functions, and communication approaches in the context of a social agent providing service or companionship to users with developmental disabilities. In this thesis, I describe an exploratory study of interaction design for a socially assistive robot that supports people suffering from developmental disabilities. While exploring the impacts of visual elements to robot's visual interface and different aspects of robot's social dimension, I developed a series of prototypes and tested them through three user studies that included three residents with various function levels at a local group home for people with developmental disabilities. All user studies had been recorded for the following qualitative data analysis. Results show that each design factor played a different role in delivering information and in increasing engagement, and there are more aspects of HRI to consider besides robot's graphical user interface and speech, such as proxemics and robot's physical appearance and dimensions. I also note that some fundamental design principles that would work for ordinary users did not apply to our target user group. I conclude that socially assistive robots could benefit our target users and acknowledge that these robots were not suitable for certain scenarios based on the feedback from our users.