Abstract: Blockchain sharding is widely considered to be one of the most effective techniques for improving blockchain performance, which divides the blockchain network into multiple shards to process transactions in parallel. However, existing sharding schemes focus on optimizing point-to-point asset transfer transactions, with limited focus on 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, designing Blockchain Sharding with Smart Contract Compatibility (BS-SCC) schemes presents significant challenges. Although there are some studies in the literature, there still lacks a thorough survey on BS-SCC. This paper provides a comprehensive survey on existing BS-SCC schemes. First, we introduce the research motivation and basic concepts of BS-SCC. Next, we provide taxonomies of BS-SCC based on transaction execution and state storage. Then, we propose a set of criteria for evaluating BS-SCC that encompass reliability, effectiveness, and other additional properties. Furthermore, we conduct a thorough review on existing BS-SCC schemes by applying our proposed criteria as a measure to justify their pros and cons. Finally, a list of open issues is highlighted, and corresponding future research directions are suggested to offer theoretical insights and practical guidance for advancing the BS-SCC.