Covering theorems are known to be among some of the fundamental tools in measure theory. They reflect the geometry of the space and are commonly used to establish connections between local and global behavior of measures. In this talk, after giving an overview of the general philosophy of covering theorems, I will more specifically turn my attention to the Besicovitch covering property (BCP) which originates from the work of Besicovitch in the 40's in connection with the theory of differentiation of measures. In the Euclidean, and Riemannian, setting, the validity of BCP is well understood. In other geometric contexts, the question whether a metric space can be remetrized so that BCP holds turns out to be much more delicate. I will illustrate this issue in the sub-Riemannian setting of the Heisenberg groups, sub-Riemannian geometry having indeed received a growing interest in recent years. Until recently, all commonly used homogeneous distances on the Heisenberg groups were known not to satisfy BCP. Surprisingly, in a recent joint work with E. Le Donne, we were able to provide a homogeneous distance on these groups for which BCP holds.