This talk, aimed at a general audience of mathematicians, discusses very geometric models of exotic spheres.

Exotic spheres are manifolds homeomorphic to classical spheres but not diffeomorphic to them. We will outline some constructions via group actions and discuss their admissible geometries. For instance, until this day, it is not known whether an exotic sphere admits a metric of positive sectional curvature, but some do not even admit positive scalar curvature.

In this talk, we also outline how one can indirectly study their geometries, aiming to gain a better understanding of their admissible geometries, for example, regarding the existence of Lorentzian metrics and the spectral problem.