The ever increasing ability of medical technology to reshape the human body in fundamental ways - from organ and tissue transplants to reconstructive surgery and prosthetics - is something now largely taken for granted. But for a philosopher, such interventions raise fundamental and fascinating questions about our sense of individual identity and its relationship to the physical body. Drawing on and engaging with philosophers from across the centuries, Jenny Slatman here develops a novel argument: that our own body always entails a strange dimension, a strangeness that enables us to incorporate radical physical changes.