Serverless computing is an approach to cloud computing that allows programmers to run serverless functions in response to external events. Serverless functions are priced at sub-second granularity, support transparent elasticity, and relieve programmers from managing the operating system. Thus serverless functions allow programmers to focus on writing application code, and the cloud provider to manage computing resources globally. Unfortunately, today's serverless platforms exhibit high latency, because it is difficult to maximize resource utilization while minimizing operating costs. This paper presents serverless function acceleration, which is an approach that transparently lowers the latency and resource utilization of a large class of serverless functions. We accomplish this using language-based sandboxing, whereas existing serverless platforms employ more expensive operating system sandboxing technologies, such as containers and virtual machines. OS-based sandboxing is compatible with more programs than language-based techniques. However, instead of ruling out any programs, we use language-based sandboxing when possible, and OS-based sandboxing if necessary. Moreover, we seamlessly transition between language-based and OS-based sandboxing by leveraging the fact that serverless functions must tolerate re-execution for fault tolerance. Therefore, when a serverless function attempts to perform an unsupported operation in the language-based sandbox, we can safely re-execute it in a container. We use a new approach to trace compilation to build source-level, interprocedural, execution trace trees for serverless functions written in JavaScript. We compile trace trees to a safe subset of Rust, validate the compiler output, and link it to a runtime system. We evaluate these techniques in our implementation, which we call Containerless.