Any advice on this subject beyond the obvious (be yourself, no typos, etc.)? What is the specific question that they want answered. Any takers?

The question above is directed towards Oxford, Cambridge, and the London School of Economics. I should have stated that above.

Any advice? I'm completely stuck here.

One piece of advice would be to not worry about it all that much.

My personal statements were pretty much boiler-plate stream-of-consciousness CV summaries that stopped when I ran out of space, and as of yet no one's come by to shatter my kneecaps.

Maybe just watch for typos, and keep the profanity to a minimum? (Dag-nabbit.)

I think Bender and I agree that you should not worry too much about it, and I don't want to take his point unfairly out of context, but I do want to add a qualification to his point.

I think you just have to make an argument for yourself. You've got a CV (and that's where Bender and I agree) but rather than argue that your qualifications justify their admitting you to the programme (which is the part that takes Bender's point unfairly out of context), I think you might also consider arguing why it makes sense for you to be motivated to take up study on the LL.M. What does their programme do for you? (That's my qualification.)

I think it was the strength of my personal statement which led to being offered a place by Bristol, as I didn't meet the entry requirements, so hopefully I can help.

I started by saying why I wanted to do the course. I went into some detail, commenting on some of the modules I would find interesting. This shows that you have tailored your personal statement to that specific institution, it impresses them and grabs their attention.

My second paragraph comments on my motivation to do the LLM - this may not be relevant to you as in this paragraph I was trying to excuse my failure to meet the entry requirements, trying to turn it into an advantage by saying that I have something to prove.

My third paragraph detailed the extra-curricular activities on offer by the university and how they would benefit me.

My fourth paragraph commented on the city in general and how it would be a good place to study.

My final paragraph discussed what I would do after the LLM career-wise, and how the LLM would be instrumental in me attaining my chosen career.

Remember to show absolute, unflinching self-belief as this will impress them also. But avoid arrogance - there is quite a fine line between the two!