Since the groundbreaking work of M. Gromov in the 1980s many tools have been developed for distinguishing open symplectic domains. However, until recently, similar questions in the contact geometric setup were largely open. For instance, it was not known whether there are open domains in the standard contact vector space of dimension $>3$ which are diffeomorphic but not contactomorphic to it (in dimension $3$ it is known that all of them are). In my lecture I will discuss Floer theoretic tools for answering this type of questions. As one application I will construct a continuous family of pairwise non-contactomorphic open balls in the standard contact ${\mathbb R}^5$. The lecture is based on a joint work in progress with K. Ajij, Mahan Mj, Dishant Pancholi and L. Polterovich.