User mobility is rapidly becoming an important and popular feature in today's networks. This is especially evident in wireless/cellular environments. While useful and desirable, user mobility raises a number of important security-related issues and concerns. One of them is the issue of tracking mobile user's movements and current whereabouts. Ideally, no entity other than the user himself and a responsible authority in the user's home domain should know either the real identity or the current location of the mobile user. At present, environments supporting user mobility either do not address the problem at all or base their solutions on the specific hardware capabilities of the user's personal device, e.g., a cellular telephone. This paper discusses a wide range of issues related to anonymity in mobile envlronments, reviews current state-of-the-art approaches and proposes several potential solutions. Solutions vary in complexity, degree of protection and assumptions about the underlying environment.