Printed circuit boards (PCBs) are a basic necessity for all modern electronic systems but are becoming increasingly vulnerable to cloning, overproduction, tampering, and unauthorized operation. Most efforts to prevent such attacks have only focused on the chip level, leaving a void for PCBs and higher levels of abstraction. In this article, we propose the first ever obfuscation-based framework for the protection of PCBs. Central to our approach is a permutation block that hides the inter-chip connections between chips on the PCB and is controlled by a key. If the correct key is applied, then the correct connections between chips are made. Otherwise, the connections are incorrectly permuted, and the PCB/system fails to operate. We propose a permutation network added to the PCB based on a Benes network that can easily be implemented in a complex programmable logic device or field-programmable gate arrays. Based on this implementation, we analyze the security of our approach with respect to (i) brute-force attempts to reverse engineer the PCB, (ii) brute-force attempts at guessing the correct key, and (iii) physical and logistic attacks by a range of adversaries. Performance evaluation results on 12 reference designs show that brute force generally requires prohibitive time to break the obfuscation. We also provide detailed requirements for countermeasures that prevent reverse engineering, unauthorized operation, and so on, for different classes of attackers.