A brief discussion is given of attempts that have been made to justify, in terms of electrostatic principles, the conjecture that surface charge on a conductor tends to infinity towards a convex sharp point and to zero towards a concave point, and it is concluded that the problem has not hitherto been solved. A new attempt is then made using potential-theoretic methods, and the problem is solved with a fair degree of generality. The limitations are of a kind familiar in this type of analysis and, roughly speaking, concern the degree of convexity of the conducting surface.