Take a random collection of points in $\mathbb{R}^d$ and construct a graph by connecting two points when their pairwise Euclidean distance is less than some threshold parameter $\delta>0$. This gives the random geometric graph, a classical model studied in stochastic geometry. We are interested in statistics of objects of this type, like the total edge length of the graph, when the underlying collection of points grows to infinity. In general, we consider functions of the underlying point collection which are defined in a ‘local’ way, i.e. points large distances apart do not interfere with each other’s local constructions. Such functions, suitably normalised, can be shown to satisfy a central limit theorem. In this talk, we will show how such central limit theorems arise and how to derive quantitative rates of convergence using the Malliavin-Stein method combined with the powerful concept of localization. This is based on joint work with Joseph Yukich.