||Video multicast over the heterogeneous Internet is a challenging undertaking. A promising solution is the use of multi-rate delivery, by which the receivers in a session can receive video data at different rates commensurate with their respective bandwidths. Numerous multi-rate schemes have been proposed in the literature; however, the optimal bandwidth adaptation policies for these schemes, especially with advanced video coding techniques, have yet to be established. This thesis presents a systematic study on the above problem for principal multi-rate video multicast schemes. We first study the optimal layering and bandwidth allocation for cumulative layered video multicast. We answer two fundamental questions. First, how bandwidth is allocated among video multicast sessions with uneven populations? Second, for a video session under a given bandwidth budget, how the layering structure for this session should be organized? Our study explores the flexible and dynamic property of advanced video encoders to meet the diverse requirements from the receivers. By using a simple 2-step decomposition of inter-session and intra-session allocation, we derive computationally efficient (polynomial time) algorithms for both inter-session and intra-session optimization problems. We then demonstrate that the above optimal bandwidth allocation can be seamlessly integrated into an end-to-end adaptation protocol called HALM (Hybrid Adaptation Layered Multicast). This protocol is fully compatible with the current best-effort Internet where FIFO drop-tail routers are widely used and the dominant traffic is the congestion-sensitive TCP traffic. Besides cumulative layering, we also study the optimal bandwidth adaptation for noncumulative layered video multicast. We show that this problem is considerably more complex than that in the cumulative layering case. We then present efficient heuristic solutions that achieve nearly optimal performance. Finally, we propose a novel proxy framework that is specifically designed for multicasting MPEG-4 video. We introduce the concept of Object Transmission Proxy (OTP). An OTP filters incoming streams using a bandwidth adaptation algorithm to meet dynamic network conditions. Multiple OTPs can partition the multicast tree into confined regions to handle the heterogeneity of the receivers. We also devise a set of novel adaptation algorithms that leverage the MPEG-4 object scalability. Simulation results demonstrate that, under the same link speed and network topology, our framework has substantial performance improvement over conventional frame-based bandwidth adaptation schemes. The improvement is particularly noticeable when video objects have uneven utilities, i.e., with different levels of importance to the receivers.