Recent research in affective robots has recognized their potential in supporting human well-being. Due to rapidly developing affective and artificial intelligence technologies, this field of research has undergone explosive expansion and advancement in recent years. In order to develop a deeper understanding of recent advancements, we present a systematic review of the past 10 years of research in affective robotics for wellbeing. In this review, we identify the domains of well-being that have been studied, the methods used to investigate affective robots for well-being, and how these have evolved over time. We also examine the evolution of the multifaceted research topic from three lenses: technical, design, and ethical. Finally, we discuss future opportunities for research based on the gaps we have identified in our review -- proposing pathways to take affective robotics from the past and present to the future. The results of our review are of interest to human-robot interaction and affective computing researchers, as well as clinicians and well-being professionals who may wish to examine and incorporate affective robotics in their practices.