I hope someone can help me...
I have rearranged all the items in a specific category by drag and drop, but I have no idea how to save the new ordering. There is no save button? Any ideas?

The ordering is automatically being saved, no need to save anything.

I managed to figure out why it wasn't saving. I had to go into the category and change the parent category item ordering. Category Item Layout> Other Layouts > Item Ordering - set to ordering.

Nice to hear that you got it sorted out.
FYI the ordering was being saved, you were just using another item ordering param.