An α-permanental random field is briefly speaking a model for a collection of non-negative integer valued random variables with positive associations. Though such models possess many appealing probabilistic properties, many statisticians seem unaware of α-permanental random fields and their potential applications. The purpose of this paper is to summarize useful probabilistic results, study stochastic constructions and simulation techniques, and discuss some examples of α-permanental random fields. This should provide a useful basis for discussing the statistical aspects in future work.