The ability to execute the test suite of a project is essential in many scenarios, e.g., to assess code quality and code coverage, to validate code changes made by developers or automated tools, and to ensure compatibility with dependencies. Despite its importance, executing the test suite of a project can be challenging in practice because different projects use different programming languages, software ecosystems, build systems, testing frameworks, and other tools. These challenges make it difficult to create a reliable, universal test execution method that works across different projects. This paper presents ExecutionAgent, an automated technique that installs arbitrary projects, configures them to run test cases, and produces project-specific scripts to reproduce the setup. Inspired by the way a human developer would address this task, our approach is a large language model-based agent that autonomously executes commands and interacts with the host system. The agent uses meta-prompting to gather guidelines on the latest technologies related to the given project, and it iteratively refines its process based on feedback from the previous steps. Our evaluation applies ExecutionAgent to 50 open-source projects that use 14 different programming languages and many different build and testing tools. The approach successfully executes the test suites of 33/55 projects, while matching the test results of ground truth test suite executions with a deviation of only 7.5\%. These results improve over the best previously available technique by 6.6x. The costs imposed by the approach are reasonable, with an execution time of 74 minutes and LLM costs of 0.16 dollars, on average per project. We envision ExecutionAgent to serve as a valuable tool for developers, automated programming tools, and researchers that need to execute tests across a wide variety of projects.