Abstract: Blockchain sharding, which divides a blockchain network into multiple shards to process transactions in parallel, is widely considered to be one of the most effective techniques for enhancing blockchain scalability and system performance. However, existing sharding schemes primarily focus on optimizing point-to-point asset transfer transactions and exhibit significant limitations in supporting Turing-complete smart contracts. Since executing the business logic based on Turing-complete smart contracts typically involves initiating multiple internal transactions to synchronize state across shard, designing the schemes of Blockchain Sharding with Smart Contract Compatibility (BS-SCC) presents significant and unique challenges. Although there are some studies in the literature, there is still a lack of a systematic survey to review the academic progress and research landscape. This paper provides a comprehensive survey on existing BS-SCC schemes to fill this research gap. First, we introduce the research motivation and basic concepts of BS-SCC. Next, we propose a taxonomy of BS-SCC schemes based on the dimensions of transaction execution and state storage, categorizing them into such classes as on-chain and off-chain execution models. Then, we propose a set of criteria for evaluating BS-SCC schemes that encompass reliability, effectiveness, and other performance properties. By employing the proposed criteria, we furthermore conduct a multi-dimensional comparative analysis of existing schemes. Finally, we identify a list of technical bottlenecks and suggest corresponding future research directions to offer theoretical insights and practical guidance for advancing BS-SCC.