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Humans and animals excel in combining information from multiple sensory modalities, controlling their complex bodies, adapting to growth, failures, or using tools. These capabilities are also highly desirable in robots. They are displayed by machines to some extent. Yet, the artificial creatures are lagging behind. The key foundation is an internal representation of the body that the agent - human, animal, or robot - has developed. The mechanisms of operation of body models in the brain are largely unknown and even less is known about how they are constructed from experience after birth. In collaboration with developmental psychologists, we conducted targeted experiments to understand how infants acquire first "sensorimotor body knowledge". These experiments inform our work in which we construct embodied computational models on humanoid robots that address the mechanisms behind learning, adaptation, and operation of multimodal body representations. At the same time, we assess which of the features of the "body in the brain" should be transferred to robots to give rise to more adaptive and resilient, self-calibrating machines. We extend traditional robot kinematic calibration focusing on self-contained approaches where no external metrology is needed: self-contact and self-observation. Problem formulation allowing to combine several ways of closing the kinematic chain simultaneously is presented, along with a calibration toolbox and experimental validation on several robot platforms. Finally, next to models of the body itself, we study peripersonal space - the space immediately surrounding the body. Again, embodied computational models are developed and subsequently, the possibility of turning these biologically inspired representations into safe human-robot collaboration is studied.