Physical embodiment is a required component for robots that are structurally coupled with their real-world environments. However, most socially interactive robots do not need to physically interact with their environments in order to perform their tasks. When and why should embodied robots be used instead of simpler and cheaper virtual agents? This paper reviews the existing work that explores the role of physical embodiment in socially interactive robots. This class consists of robots that are not only capable of engaging in social interaction with humans, but are using primarily their social capabilities to perform their desired functions. Socially interactive robots provide entertainment, information, and/or assistance; this last category is typically encompassed by socially assistive robotics. In all cases, such robots can achieve their primary functions without performing functional physical work. To comprehensively evaluate the existing body of work on embodiment, we first review work from established related fields including psychology, philosophy, and sociology. We then systematically review 65 studies evaluating aspects of embodiment published from 2003 to 2017 in major peer-reviewed robotics publication venues. We examine relevant aspects of the selected studies, focusing on the embodiments compared, tasks evaluated, social roles of robots, and measurements. We introduce three taxonomies for the types of robot embodiment, robot social roles, and human-robot tasks. These taxonomies are used to deconstruct the design and interaction spaces of socially interactive robots and facilitate analysis and discussion of the reviewed studies. We use this newly-defined methodology to critically discuss existing works, revealing topics within embodiment research for social interaction, assistive robotics, and service robotics, in which more extensive exploration would greatly improve the current understanding of the impact of embodiment on human perception and evaluation of human-robot interactions. The introduced taxonomy for embodiment design is used as a starting point for outlining our characterization of the design space of robot embodiments. The presented characterization can be used to discuss how the physical embodiment of socially interactive robots relates to social capabilities and affordances. By introducing a general model of the design space, existing research findings can better advise robot designers and we discuss how these findings can inform researchers through design decisions in the development of future socially interactive robots.