A little over a decade ago, Taubes and his disciples started to observe non-compactness phenomena in gauge theories in dimensions three and four that are linked with multi-valued harmonic spinors, 1–forms, self-dual 2–forms, etc. This can be understood as a higher-dimensional shadow of the Hitchin map from the moduli space of Higgs bundles to the space of quadratic differentials. The purpose of this talk is to motivate the study of multi-valued harmonic spinors, provide a survey of some of the advances that have been made, and discuss some open problems. In particular, I will touch up on recent joint work with Bera and work in progress with He and Parker regarding the deformation theory of multivalued harmonic spinors. This talk is part of the activities of the FIS2 project BeyondCY3 - CUP B53C25001070001. More information available at: https://sites.google.com/uniroma1.it/lfoscolo/fis2-beyondcy3/events