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In this paper, we propose a novel interference-suppressed semantic-aware non-orthogonal multiple access (IS-SNOMA) framework for downlink image transmission in the semantic-aware communication networks. The proposed IS-SNOMA is able to mitigate the inter-user interference in the non-orthogonal transmission for multiple semantic-oriented users (SU) or the coexistence of SUs and bit-oriented users (BU). 1) For the homogeneous transmission of semantic streams, we propose a pair of novel semantic difference (SeD) aware IS-SNOMA transceivers to accommodate multiple SUs over the same resource block. A novel SeD-aware superposition coding (SeDSC) technique and SeD-aware successive interference cancellation (SeDSIC) technique are specially designed to mitigate the semantic-level interference. 2) For the heterogeneous transmission of semantic-bit streams, we develop a pair of syntactic …