Gauge theories have provided our best representations of the fundamental forces of nature. But a representation may be successfully employed even without a clear understanding of how it works. This book seeks such understanding, so that we are able to say what kind of world our gauge theories reveal to us. A gauge theory's representations are mathematical structures that may be transformed while certain features remain the same. Do representations related by such a gauge transformation merely offer alternative perspectives on the same situation? If so, then gauge symmetry is a purely formal property since it reflects no corresponding symmetry in nature. This book describes the representations provided by gauge theories in both classical and quantum physics. It argues that gauge symmetry is a purely formal property of almost all classes of representations these provide. Evidence for classical gauge theories of forces (other than gravity) gives us reason to believe that loops rather than points are the locations of fundamental properties. Besides exploring whether this holds also for the quantum gauge field theories of the Standard Model of elementary particle physics, the book assesses the difficulties involved in basing such ontological conclusions on the success of these theories.