Zoom-Meeting: https://wwu.zoom.us/j/97461739978 I will report on a recent joint work with J. Carlos Díaz-Ramos and Alberto Rodríguez-Vázquez where we conclude the classification of cohomogeneity one actions on symmetric spaces of rank one (up to orbit equivalence) by classifying such actions on quaternionic hyperbolic spaces. Moreover, as a by-product of our approach, we find uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces. This provides a striking contrast to the isoparametric theory in the other symmetric spaces of rank one.