Many fields of pure and applied mathematics deal with results of classification for topological structures, which are highly symmetric. While the presence of a symmetry can be detected at the level of commuting elements, it is sometime possible to look for different symmetries which involve commuting substructures more than commuting elements. This intuition was formalised by Sir William Rowan Hamilton in 1843 and had profound consequences in pure mathematics and theoretical physics. His celebrated quaternion group of order eight was the first example of a finite nonabelian group, where any randomly picked pair of subgroups is commuting. During the successive years, many intuitions of Hamilton were refined and adapted to different areas of pure and applied mathematics. The present talk will survey some new ideas on topologically quasihamiltonian groups, referring to a recent monograph on the topic. Time permitting, open problems will be presented both in the area of topology and in the area of dynamical systems.