Smart contract, one of the most successful applications of blockchain, provides the foundation for realizing various real-world applications of blockchain, playing an essential role in the blockchain ecosystem. However, frequent smart contract security events not only caused huge economic losses but also destroyed the blockchain-based credit system. The security and reliability of smart contract thus gain wide attention from researchers worldwide. This study first introduces the common types and typical cases of smart contract vulnerabilities from three levels, i.e., Solidity code layer, EVM execution layer, and blockchain system layer. Then, the research progress of smart contract vulnerability detection is reviewed and existing efforts are classified into five categories, namely formal verification, symbolic execution, fuzzing testing, intermediate representation, and deep learning. The detectable vulnerability types, accuracy, and time consumption of existing vulnerability detection methods are compared in detail as well as their limitations and improvements. Finally, based on the summary of existing researches, the challenges in the field of smart contract vulnerability detection are discussed and combined with the deep learning technology to look forward to future research directions.