We study the diameter of scale-free networks, having degree distribution P(k) ~ k^-a (i.e. the probability of having k links outgoing from a site). We show that for a>2 a lower bound for the diameter of the network is d ~ ln ln N, where N is the number of sites in the network. While the diameter of regular random networks or small world networks is known to be d ~ ln N, we present plausible arguments that scale free networks with 2<a3, d ~ ln N.