The iOS Persistence and Core Data course on Udacity covers various topics related to data persistence in iOS apps. The course begins with an introduction to simple persistence, teaching students how to save small pieces of data and set user preferences using NSUserDefaults. It then moves on to discussing the iOS File System and the app's sandbox, teaching students how to read and save files using NSFileManager and understand the performance implications of file operations. The course then introduces Core Data, Apple's framework for managing the data layer, explaining what a data layer is and guiding students through converting a non-Core Data app to have a Core Data model. Next, the course covers the Core Data stack, teaching students how to set up the necessary classes and use the stack to manage model object creation and deletion, as well as persist changes. The course then focuses on simplifying code using Core Data, enabling user interfaces to reactively update when the model changes, setting up an NSFetchedResultsController to observe data changes and notify the UI, and creating a new version of a data model, modifying a table view to work with a fetched results controller, and enabling caching. Finally, the course covers Core Data and concurrency, teaching students how to update the data model and safely migrate user data between versions, work with multiple managed object contexts for different tasks, and keep the user interface responsive by sending lengthy tasks to a background queue.