In a recent joint paper with Misha Belolipetsky, Sasha Kolpakov and Leone Slavich we developed a large industry connecting geometry and arithmetic of hyperbolic orbifolds and manifolds. We introduce a new class of the so-called finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic orbifolds: a hyperbolic orbifold M is arithmetic if and only if it has infinitely many fc-subspaces. We also show that immersed totally geodesic m-dimensional suborbifolds of n-dimensional arithmetic hyperbolic orbifolds are fc-subspaces whenever \(m\geq \lfloor n/2\rfloor\), and we provide examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. One of the key results of our paper is a characterization of immersions of arithmetic hyperbolic orbifolds into each other. In the first part of the talk I will give an overview of this area and present our main results, and the second part (it time permits) will be devoted to our methods.