Ensuring secure access control to multicast delivery trees is a challenging issue that is still largely open. Moreover, the impact of host mobility on the access control to the delivery tree has not been investigated. In this article we address the receiver and sender access control problems to the delivery tree and extend the interest to the mobile IP environment by taking into account its particular issues and requirements. To achieve this, we review the existing approaches by classifying them into three classes: digital signature-based solutions, shared secret-based solutions, and hybrid solutions. We also investigate their efficiency and limitations with respect to the specified requirements both in stationary and mobile cases. Our study shows that four main problems arise among the existing approaches. First, few approaches addressed the sender access control problem. Second, both the digital signature- based solutions and hybrid solutions are vulnerable to DoS attacks. Third, the existing solutions do not provide an efficient user exclusion mechanism. Fourth, the defined access control mechanisms result in a number of problems in mobile IP environments.