On Retellings

So here's the question of the day: What do you think about retellings?

For me, I like a retelling that stays true to the story. I don't mind a twist, I don't mind changes. I don't mind if certain characters are cut or if scenes or plot points from the original are missing. I don't mind if a classic becomes an epic fantasy story of if a fairy tale turns sci-fi. But I like my retellings to stick to what makes the original good. It's taken me some time to learn that staying too close to the original isn't a good thing to do, but I'm not a fan when retellings go completely another way until it's really not much of a retelling at all. But that's just my personal opinion.
Some of my favorite retellings
What are your thoughts? For example, what if the ending is completely different? What if one (or more) of the characters is changed in a fundamental way? I think it's safe to say that setting can be a total free-for-all, but again, maybe this is just my personal opinion and someone else might not like it if the setting is way out there from the original.
Shakespeare Retold: Much Ado About Nothing (part one)
I watched this recently. The end was different from the original but made total sense for this retelling.

What kinds of things stay the same and what changes in your favorite retellings? Is everything about the story safe for the author to change, or should some things automatically be included? I would love to hear your thoughts/opinions.