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With scalable video coding that provides fine-granular quality degradation, such as fine granularity scalability (FGS) and progressive FGS (PFGS), or H.264 scalable video coding's (SVC) adaptive reference FGS (AR-FGS) coding, video can flexibly be streamed to receivers of heterogeneous bandwidths. However, the transmitted video is only efficiently encoded when the transmission bit rate is in the vicinity of the encoding bit rate. In this paper, we develop and evaluate a comprehensive suite of network-aware adaptive bitstream switching policies for point-to-point and point-to-multipoint streaming of fine granular scalable coded video to address this coding efficiency issue. Our approach stores a small number of encodings (versions) with different encoding bit rates for each video sequence and estimates the reconstructed quality using the motion activity levels of the underlying visual content (or, in general, any content descriptor(s) that highly correlate with the reconstructed quality). For unicast streaming, we then: (i) adaptively switch between the different encodings at the server, to improve the reconstructed video quality and (ii) adaptively drop packets during network congestion to ensure fairness between multiple unicast streams. For multicast streaming, we also adaptively switch between the different encodings to maximize the average video quality. Our adaptive bitstream switching policies consider the visual content descriptors as well as the network channel variability, while requiring only sample points from the rate-distortion curve of the video stream. From our extensive simulations with PFGS coding, we find that our adaptive unicast bitstream switching policy achieves on average a 0.8dB improvement over the optimal non-adaptive streaming for a diverse 200-shot sequence from Star Wars IV. We have also verified our key findings with the latest scalable video coding standard, H.264 SVC.