We present an embodied theory of transfer as applied to mathematical ideas. Project-based learning (PBL) offers important sites for investigating transfer because students and teachers must track mathematical invariant relations that are embedded in complex settings where knowledge and information is distributed and extended across various materials, symbolic inscriptions, and social interactions. We argue that transfer occurs when learners and teachers establish cohesion of their experiences by mapping modes of perceiving and acting that they successfully used in previous contexts to new contexts. Learners express that cohesion across contexts in a variety of ways, principally through speech, actions, and gestures, including gestural catchments and simulated actions. Students and teachers may engage in several forms of mapping, including explicit analogical mapping, constructing mappings implicitly via relational priming, and mapping relations via conceptual blending. Thus, we theorize that both teachers and students are integral to the transfer process. This embodied theory of transfer can account for near and far transfer, negative transfer, and "false" transfer, in which actors enact modes of perceiving and acting activated by cues that match only at a surface level. Embodied accounts of transfer have implications for educational practice. Some pedagogical approaches, such as concreteness fading, aim to foster and maintain cohesion of mathematical invariant relations. An embodied perspective on transfer can also inform the design of assessments, including both formative and summative assessments. [This chapter was published in: Hohensee, C., Lobato, J. (Eds.), "Transfer of Learning Progressive Perspectives for Mathematics Education and Related Fields," Research in Mathematics Education, (pp. 27-58). Chaim, Switzerland: Springer.]