Nowadays, enterprises widely deploy Network Functions (NFs) and server applications in the cloud. However, processing of sensitive data and trusted execution cannot be securely deployed in the untrusted cloud. Cloud providers themselves could accidentally leak private information (e.g., due to misconfigurations) or rogue users could exploit vulnerabilities of the providers' systems to compromise execution integrity, posing a threat to the confidentiality of internal enterprise and customer data. In this paper, we identify (i) a number of NF and server application use-cases that trusted execution can be applied to, (ii) the assets and impact of compromising the private data and execution integrity of each use-case, and (iii) we leverage Intel's Software Guard Extensions (SGX) architecture to design Trusted Execution Environments (TEEs) for cloud-based NFs and server applications. We combine SGX with the Data Plane Development KIT (DPDK) to prototype and evaluate our TEEs for a number of application scenarios (Layer 2 frame and Layer 3 packet processing for plain and encrypted traffic, traffic load-balancing and back-end server processing). Our results indicate that NFs involving plain traffic can achieve almost native performance (e.g., ~22 Million Packets Per Second for Layer 3 forwarding for 64-byte frames), while NFs involving encrypted traffic and server processing can still achieve competitive performance (e.g., ~12 Million Packets Per Second for server processing for 64-byte frames).