In the first part of this paper, under the assumption of noiseless transmission, we develop two entropy-coded subband image coding schemes. The difference between these schemes is the procedure used for encoding the lowest frequency subband: predictive coding is used in one system and transform coding in the other. Other subbands are encoded using zero-memory quantization. After a careful study of subband statistics, the quantization parameters the corresponding Huffman codes and the bit allocation among subbands are all optimized. It is shown that both schemes perform considerably better than the scheme developed by Woods and O'Neil [2]. Roughly speaking, these new schemes perform the same as that in [2] at half the encoding rate. In the second part of the paper, after demonstrating the unacceptable sensitivity of these schemes to transmission noise, we will develop a combined source/channel coding scheme in which rate-compatible convolutional codes are used to provide protection agains channel noise. A packetization scheme to prevent infinite error propagation is used and an algorithm for optimal assignment of bits between the source and channel encoders of different subbands is developed. We will show the in the presence of channel noise, these channeloptimized schemes offer dramatic performance improvements ove' the schemes designed based on a noiseless channel assumption; they also perform better than that of [2] even in th absence of channel noise. Finally, the robustness of the proposed schemes against channel mismatch will be studied.