Abstract. It is known that for each d there exists a graph of diameter two and maximum degree d which has at least |_(d+1)/2_||_(d+3)/2_| vertices. In contrast with this, we prove that for every surface S there is a constant dS such that each graph of diameter two and maximum degree d>dS, which is embeddable in S, has at most |_(3d)/2_|+1 vertices. This upper bound is best possible, and we show that extremal graphs can be found among surface triangulations.