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We study the cover time of random geometric graphs. Let I(d) = [0, 1]d denote the unit torus in d dimensions. Let D(x, r) denote the ball (disc) of radius r. Let ϒd be the volume of the unit ball D(0, 1) in d dimensions. A random geometric graph G = G(d, r, n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(x, r) of radius r about x. The vertex set V(G) = V and the edge set E(G) = {{v, w}: w ≠ v, w ∈ D(v, r)}. Let G(d, r, n), d ≥ 3 be a random geometric graph. Let c 1 be constant, and let r = (c log n/(ϒdn))1/d. Then whp CG ~ clog(c/c-1)nlogn.