Anonymity on the Internet is a property commonly identified with privacy of electronic communications. A number of different systems exist which claim to provide anonymous email and web browsing, but their effectiveness has hardly been evaluated in practice. In this thesis we focus on the anonymity properties of such systems. First, we show how the anonymity of anonymity systems can be quantified, pointing out flaws with existing metrics and proposing our own. In the process we distinguish the anonymity of a message and that of an anonymity system. Secondly, we focus on the properties of building blocks of mix-based (email) anonymity systems, evaluating their resistance to powerful blending attacks, their delay, their anonymity under normal conditions and other properties. This leads us to methods of computing anonymity for a particular class of mixes – timed mixes – and a new binomial mix. Next, we look at the anonymity of a message going through an entire anonymity system based on a mix network architecture. We construct a semantics of a network with threshold mixes, define the information observable by an attacker, and give a principled definition of the anonymity of a message going through such a network. We then consider low latency connection-based anonymity systems, giving concrete attacks and describing methods of protection against them. In particular, we show that Peer-to-Peer anonymity systems provide less anonymity against the global passive adversary than ones based on a “classic” architecture. Finally, we give an account of how anonymity can be used in censorship resistant systems. These are designed to provide availability of documents, while facing threats from a powerful adversary. We show how anonymity can be used to hide the identity of the servers where each of the documents are stored, thus making them harder to remove from the system.