It has long been thought that macroscopic phase coherence breaks down in effectively lower-dimensional superconducting systems even at zero temperature due to enhanced topological quantum phase fluctuations. In quasi-one-dimensional wires, these fluctuations are described in terms of 'quantum phase-slip' (QPS): tunneling of the superconducting order parameter for the wire between states differing by plus or minus 2 pi in their relative phase between the wire's ends. Over the last several decades, many deviations from conventional bulk superconducting behavior have been observed in ultra-narrow superconducting nanowires, some of which have been identified with QPS. While at least some of the observations are consistent with existing theories for QPS, other observations in many cases point to contradictory conclusions or cannot be explained by these theories. Hence, our understanding of the nature of QPS, and its relationship to the various observations, has remained incomplete. In this paper we present a new model for QPS which takes as its starting point an idea originally postulated by Mooij and Nazarov (2006 Nature Phys. 2 169): that flux-charge duality, a classical symmertry of Maxwell's equations, can be used to relate QPS to the well-known Josephson tunneling of Cooper pairs. Our model provides an alternative, and qualitatively different, conceptual basis for QPS and the phenomena which arise from it in experiments, and it appears to permit for the first time a unified understanding of observations across several different types of experiments and materials systems.