Debbie
You can only delete a question, old or new, that doesn't have any answers. Once a question gets its first answer, the option to delete is removed so then the only way it can be deleted is if it's reported as a violation by other users. If it has no answers you can click on the blue edit button and choose the delete option from there. In the meantime, you can use the "make anonymous" feature found in your edit button so that at least your username and avatar are removed from it, this is most helpful if your question is embarrassing in some way.
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Questions can now be answered forever even with a best answer. You cannot stop that and you also cannot delete questions that have even just 1 answer.
Before if the best answer was a violation and deleted the question went away too but now the question stays posted and the asker can choose a new best answer.
TheHumbleOne
I have a couple of questions which are quite old yet I still get answers. How can I delete them or indicate that they have been answered?