Serverless computing is a relatively recent cloud computing paradigm that allows customers to write lightweight, short-lived functions that are executed on resources provisioned and managed by cloud providers. This architecture was originally designed to simplify stateless, event-driven applications such as those designed to handle web requests, compress images, or generate thumbnails. However, this paradigm has been increasingly adopted in other domains including data analytics, machine learning, and sensor data processing. These domains, and other potential applications, could benefit from better fault-tolerance, consistent concurrent file access, and improved I/O performance than are provided by current serverless offerings. In this paper we propose Savanna, a system that implements these features for serverless applications. Savanna requires applications to use a file API, but is otherwise transparent to serverless applications.