The introduction of collaborative robots aims to make production more flexible, promoting a greater interaction between humans and robots also from physical point of view. However, working closely with a robot may lead to the creation of stressful situations for the operator, which can negatively affect task performance. In Human-Robot Interaction (HRI), robots are expected to be socially intelligent, that is, capable of understanding and reacting accordingly to human social and affective clues. This ability can be exploited implementing affective computing, which concerns the development of systems able to recognize, interpret, process, and simulate human affects. Social intelligence is essential for robots to establish a natural interaction with people in several contexts, including the manufacturing sector with the emergence of Industry 5.0. In order to take full advantage of the human-robot collaboration, the robotic system should be able to perceive the psycho-emotional and mental state of the operator through different sensing modalities (e.g. facial expressions, body language, voice, or physiological signals) and to adapt its behavior accordingly. The development of socially intelligent collaborative robots in the manufacturing sector can lead to a symbiotic human-robot collaboration, arising several research challenges that still need to be addressed. The goals of this paper are the following: (i) providing an overview of affective computing implementation in HRI; (ii) analyzing the state-of-art on this topic in different application contexts (e.g. healthcare, service applications, and manufacturing); (iii) highlighting research challenges for the manufacturing sector.