In modern networks, forwarding of packets often depends on the history of previously transmitted traffic. Such networks contain stateful middleboxes, whose forwarding behavior depends on a mutable internal state. Firewalls and load balancers are typical examples of stateful middleboxes. This paper addresses the complexity of verifying safety properties, such as isolation, in networks with finite-state middleboxes. Unfortunately, we show that even in the absence of forwarding loops, reasoning about such networks is undecidable due to interactions between middleboxes connected by unbounded ordered channels. We therefore abstract away channel ordering. This abstraction is sound for safety, and makes the problem decidable. Specifically, we show that safety checking is EXPSPACE-complete in the number of hosts and middleboxes in the network. We further identify two useful subclasses of finite-state middleboxes which admit better complexities. The simplest class includes, e.g., firewalls and permits polynomial-time verification. The second class includes, e.g., cache servers and learning switches, and makes the safety problem coNP-complete. Finally, we implement a tool for verifying the correctness of stateful networks.