Movement-based interactive systems for play came into the spotlight over a decade ago, and were met with enthusiasm by the general public as well as the Human-Computer Interaction research community. Yet a decade of research and practice has not fully addressed the challenge of designing for the moving body and play. This thesis argues that often, the role of the technology to sustain the play activity, and to drive the design process, has been over-emphasized, and has resulted in limited design possibilities. This thesis explores an alternative design approach to address the problem through combining the design of the technology with designing aspects of the social and spatial context where the play activity takes place. The work is grounded in an embodied perspective of experience, action, and design. Methodologically, it belongs to the Research through Design tradition (RtD). A core concept and a characterization of design practices are presented as key contributions. The concept of embodied core mechanics is introduced to frame desirable and repeatable movement-based play actions, paying attention to the way these are supported by design resources including rules, physical and digital artifacts, and the physical and spatial arrangement of players and artifacts. The concept was developed during the two main design cases: the Oriboo case, targeting dance games for children, and the PhySeEar case, targeting rehabilitative therapy for the elderly. It was further substantiated in subsequent external design collaborations. To support the design process, this thesis presents embodied sketching: a set of ideation design practices that leverage the embodied experience and enable designers to scrutinize the desired embodied experience early in the design process. Three forms of embodied sketching are presented: embodied sketching for bodystorming, co-designing with users, and sensitizing designers. Through reframing the design task as one of designing and studying embodied core mechanics, this thesis establishes an alternative approach to design for movement-based play in which significant aspects of the embodied play experience, lead, drive, and shape the design process and the design of the technology.