The releases of Intel SGX and AMD SEV mark the transition of hardware-based enclaves from research prototypes to mainstream products. These two paradigms of secure enclaves are attractive to both the cloud providers and tenants, since security is one of the key pillars of cloud computing. However, it is found that current hardware-defined enclaves are not flexible and efficient enough for the cloud. For example, although SGX can provide strong memory protection with both confidentiality and integrity, the size of secure memory is tightly restricted. On the contrary, SEV enables enclaves to use more memory but has critical security flaws due to no memory integrity protection. Meanwhile, both types of enclaves have relatively long booting latency, which makes them not suitable for short-term tasks like serverless workloads. After an in-depth analysis, we find that there are some intrinsic tradeoffs between security and performance due to the limitation of architectural designs. In this article, we investigate a novel hardware-software co-design of enclaves to meet the requirements of cloud by placing a part of the logic of the enclave mechanism into a lightweight software layer, named Enclavisor, to achieve a balance between security, performance, and flexibility. Specifically, our implementation is based on AMD's SEV and, Enclavisor is placed in the guest kernel mode of SEV's secure virtual machines. Enclavisor inherently supports memory encryption with no memory limitation and also achieves efficient booting, multiple enclave granularities, and post-launch remote attestation. Meanwhile, we also propose hardware/software solutions to mitigate the security flaws caused by the lack of memory integrity. We implement a prototype of Enclavisor on an AMD SEV server. The experiments on both micro-benchmarks and application benchmarks show that enclaves on Enclavisor can have close-to-native performance.