Does every countably infinite group G have a Cayley graph of infinite diameter? While there are examples of uncountable groups that fail to satisfy this property (e.g., the group of all permutations of the integers), the question for countable groups remains open. After reviewing the necessary background and some known results, I will discuss an attempt to solve this problem by choosing a random generating set. For a wide class of countable groups, this approach answers the question affirmatively and reveals a surprising phenomenon: random generating sets yield the same Cayley graph, independent of the group! Depending on the randomness model, this graph is either the familiar Rado graph (which has diameter 2) or a certain mysterious graph of infinite diameter