In Human-Robot Interactions (HRI), robots should be socially intelligent. They should be able to respond appropriately to human affective and social cues in order to effectively engage in bi-directional communications. Social intelligence would allow a robot to relate to, understand, and interact and share information with people in real-world human-centered environments. This survey paper presents an encompassing review of existing automated affect recognition and classification systems for social robots engaged in various HRI settings. Human-affect detection from facial expressions, body language, voice, and physiological signals are investigated, as well as from a combination of the aforementioned modes. The automated systems are described by their corresponding robotic and HRI applications, the sensors they employ, and the feature detection techniques and affect classification strategies utilized. This paper also discusses pertinent future research directions for promoting the development of socially intelligent robots capable of recognizing, classifying and responding to human affective states during real-time HRI.