This paper is concerned with establishing Poisson approximations to departure processes in general queueing networks. Bounds are provided to assess the degree of deviation of a given flow from Poisson and these lead to simple criteria for good Poisson approximations. The class of networks considered here are those with a countable collection of customer classes and where the service requirement of a customer at a given queue has a general distribution which may depend upon the class of the customer. Initially we consider networks of quasireversible queues and later relax this assumption