Sharding is one of the most effective technologies for improving blockchain scalability by dividing a blockchain network into multiple shards that process transactions in parallel. However, many existing sharded blockchain systems can include a large number of cross-shard transactions, resulting in high communication overhead and processing latency. Two solutions have emerged to address these issues. One of them focuses on optimizing the allocation of transactions to reduce the cross-shard communication while ensuring workload balance. The other emphasizes the design of an efficient cross-shard transaction processing mechanism to reduce the latency caused by confirming cross-shard transactions. However, the literature still lacks a comprehensive review on the advances of Cross-shard Transaction Allocation Optimization and Processing (CTAOP) mechanisms, which would be a valuable addition to the literature. This paper is a comprehensive survey of existing CTAOP mechanisms, and first introduces cross-shard transactions and related concepts. Then, we propose two sets of criteria for evaluating the two types of CTAOP mechanisms, and provide a taxonomy of CTAOP mechanisms, followed by a thorough review based on our proposed criteria. A list of open issues is also highlighted, and corresponding future research directions are suggested to advance our understanding of CTAOP mechanisms.