User mobility is rapidly becoming an important and popular network feature. This is especially evident in wireless/cellular networks where user mobility raises a number of important security issues and concerns. Foremost among them is the ability to track mobile users' movements and whereabouts. Ideally, no entity other than the user himself and a responsible authority in the user's home domain (if any) should know both the real identity and the current location of the mobile user. At present, most environments supporting user mobility either do not address the problem at all or base their solutions on assumptions that are specific to today's cellular phone networks. This paper discusses a number of issues related to anonymity and location privacy in mobile networks. It reviews current state-of-the-art approaches, identifies their exposures of anonymity and proposes several low-cost solutions which vary in complexity, degree of protection and assumptions about the underlying environment.