In distributed computing, questions of computability are exquisitely sensitive to minute details of the model assumptions, and there is no universally agreed upon model of network computing. Here, we study which functions are computable by deterministic and anonymous agents in either static or dynamic networks. We consider various communication assumptions common in the literature, and in each case we strive to characterize the set of computable functions, organizing existing results as well as offering new ones, alongside new proofs which bring new understanding of this computability landscape.