The Steam Early Access version* that I have doesn't allow manual saving, and only allows automatic saving. Due to that, it would be nice to know what actions or events will cause the game to auto-save.
What actions or events will trigger an auto-save?
* I'm making this question platform-agnostic. Answers for both the Xbox 360 and PC versions are welcome. If possible, please specify on which platform(s) your answer is about. Also, see this question's comments.