We develop techniques for measuring the amount of anonymity provided by anonymity systems in the aftermath of attacks. Anonymity systems are special systems that enable their users to carry out anonymous communication over the Internet. Our measurement techniques help compare the quality or e ectiveness of di erent anonymity systems, or even strengths of di erent attacks. We consider two types of attacks on anonymity systems, namely infeasibility and probabilistic attacks, and develop a canonical attack mapping to show that the latter is a generalization of the former. We also expand the system models underlying existing methods by including in our study emerging system features, such as data caching, message multiplicities, and their combined e ects on anonymity. Our resulting framework can be used as a basis for developing anonymity metrics for next-generation anonymity systems.