Given a point process, in Euclidean space or on a lattice, the corresponding k-point correlation function, for k=1,2, . . ., expresses the probability of finding particles at k specified points. Here we ask a converse question: if we are given a finite number of candidate correlation functions, say those for k=1,2, . . . , n, does there exist a point process which realizes these correlations? We give some partial answers to this question and discuss some examples.