An α-permanental random field is briefly speaking a model for a collection of random variables with positive associations, where α is a positive number and the probability generating function is given in terms of a covariance or more general function so that density and moment expressions are given by certain α-permanents. 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 first to summarize useful probabilistic results using the simplest possible setting, and second to study stochastic constructions and simulation techniques, which should provide a useful basis for discussing the statistical aspects in future work. The paper also discusses some examples of α-permanental random fields.