Authentication protocols are the basis of security in many distributed systems, and it is therefore essential to ensure that these protocols function correctly. Unfortunately, their design has been extremely error prone. Most of the protocols found in the literature contain redundancies or security flaws. A simple logic has allowed us to describe the beliefs of trustworthy parties involved in authentication protocols and the evolution of these beliefs as a consequence of communication. We have been able to explain a variety of authentication protocols formally, to discover subtleties and errors in them, and to suggest improvements. In this paper we present the logic and then give the results of our analysis of four published protocols, chosen either because of their practical importance or because they serve to illustrate our method.