We discuss necessary conditions for the existence of a probability distribution on particle configurations in d-dimensions, i.e., a point process, compatible with a specified density rho and radial distribution function g(r). In d = 1 we give necessary and sufficient criteria on rhog(r) for the existence of such a point process of renewal (Markov) type. We prove that these conditions are satisfied for the case g(r) = 0, r < D and g(r) = 1, r > D, if and only if rhoD less than or equal to e(-1): the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for rhog(r) to specify a determinantal point process for which all n-particle densities, rho(n)(r(1),...,r(n)), are given explicitly as determinants. We give several examples.