Mobile applications are constrained by the available memory of mobile devices. We present C-RAM, a system that uses cloud-based memory to extend the memory of mobile devices. It splits application state and its associated computation between a mobile device and a cloud node to allow applications to consume more memory, while minimising the performance impact. C-RAM thus enables developers to realise new applications or port legacy desktop applications with a large memory footprint to mobile platforms without explicitly designing them to account for memory limitations. To handle network failures with partitioned application state, C-RAM uses a new snapshot-based fault tolerance mechanism in which changes to remote memory objects are periodically backed up to the device. After failure, or when network usage exceeds a given limit, the device rolls back execution to continue from the last snapshot. C-RAM supports local execution with an application state that exceeds the available device memory through a user-level virtual memory: objects are loaded on-demand from snapshots in flash memory. Our C-RAM prototype supports Objective-C applications on the unmodified iOS platform. With C-RAM, applications can consume 10× more memory than the device capacity, with a negligible impact on application performance. In some cases, C-RAM even achieves a significant speed-up in execution time (up to 9.7×).