Large language models (LLMs) have brought significant advancements to code generation and code repair, benefiting both novice and experienced developers. However, their training using unsanitized data from open-source repositories, like GitHub, raises the risk of inadvertently propagating security vulnerabilities. Despite numerous studies investigating the safety of code LLMs, there remains a gap in comprehensively addressing their security features. In this work, we aim to present a comprehensive study aimed at precisely evaluating and enhancing the security aspects of code LLMs. To support our research, we introduce CodeSecEval, a meticulously curated dataset designed to address 44 critical vulnerability types with 180 distinct samples. CodeSecEval serves as the foundation for the automatic evaluation of code models in two crucial tasks: code generation and code repair, with a strong emphasis on security. Our experimental results reveal that current models frequently overlook security issues during both code generation and repair processes, resulting in the creation of vulnerable code. In response, we propose different strategies that leverage vulnerability-aware information and insecure code explanations to mitigate these security vulnerabilities. Furthermore, our findings highlight that certain vulnerability types particularly challenge model performance, influencing their effectiveness in real-world applications. Based on these findings, we believe our study will have a positive impact on the software engineering community, inspiring the development of improved methods for training and utilizing LLMs, thereby leading to safer and more trustworthy model deployment.