With the development of blockchain technology, the types of blockchain digital assets show a diversified trend. However, the problem of value islands in blockchain is becoming more and more prominent. Cross-chain technology serves as a bridge to realize the interconnection between blockchains, and many cross-chain projects have already realized cross-chain circulation of digital assets based on different mechanisms. This survey classifies and summarizes the diversified digital asset types, and discusses the specific needs of digital assets cross-chain in light of asset characteristics. On this basis, this survey also introduces current mainstream cross-chain mechanisms, including notary scheme, sidechain/relay, and hash-locking, and summarizes representative digital assets cross-chain projects in recent years, introduces their principles, and compares them. Finally, this survey summarizes the current status of development from the perspective of digital assets cross-chain and looks forward to future research directions.