Engineers fine-tune the design of robot bodies for control purposes; however, a methodology or set of tools is largely absent, and optimization of morphology (shape, material properties of robot bodies, etc.) is lagging behind the development of controllers. This has become even more prominent with the advent of compliant, deformable or ‘soft’ bodies. These carry substantial potential regarding their exploitation for control—sometimes referred to as ‘morphological computation’. In this article, we briefly review different notions of computation by physical systems and propose the dynamical systems framework as the most useful in the context of describing and eventually designing the interactions of controllers and bodies. Then, we look at the pros and cons of simple versus complex bodies, critically reviewing the attractive notion of ‘soft’ bodies automatically taking over control tasks. We address another key dimension of the design space—whether model-based control should be used and to what extent it is feasible to develop faithful models for different morphologies.