Can I be admitted to graduate school in a different field from my degree? Specific cases include:
If I’ve taken plenty of advanced courses in field X in the process of completing a degree in another field, can I apply to graduate school in X?
What if I haven’t taken many courses in X, but I have acquired a good grasp of X through self-study or working in a related field?
What if I’ve never studied X, but I have done very well in an unrelated field? Could I be admitted to graduate school in X on the basis of general intellectual promise, and then make up the missing background after enrollment?
Note that this question is an attempt to provide a comprehensive answer, to avoid the need for a profusion of field-specific questions on this topic (see the associated meta question). Please feel free to edit the question or answer to improve them.