For classical random graphs, the vertex set is fixed and finite, and the edges are inserted randomly. For random geometric graphs, the vertices are randomly located in space, and the edges are then inserted using some deterministic rule. A simple example is the Gilbert model, in which two vertices are joined if they lie within distance r of each other. For many such graphs, there is also a related coverage process: for instance, corresponding to the Gilbert model, we place a solid ball of radius r/2 around each randomly placed vertex. I'll discuss some recent results on such coverage processes, and on percolation in the underlying random graphs. This is joint work with Paul Balister, Bela Bollobas, Martin Haenggi and Mark Walters.