In a proximity region graph G in R d , two distinct points x, y of a point process µ are connected when the 'forbidden region' S(x, y) these points determine has empty intersection with µ. The Gabriel graph, where S(x, y) is the open disc with diameter the line segment connecting x and y, is one canonical example. Under broad conditions on the process µ and regions S(x, y), bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of G, including the total number of edges, and total length.