@Jo - "Don't wish to be anal, but I would say that BJ's example "I'm quitting as of Friday" was a bit off point. [Instead use:]

I'm quitting on Friday
or
As of Friday I will no longer be working for 'XYZ Ltd'"

For what it's worth, I agree with your correction Jo - my example was a bit off because the act of quitting is something that occurs at a particular point in time. Your second example makes more sense; or to preserve the structure of my example, something like:

"I will be looking for a new job as of Friday."

I continue to share your view that in BrE, "as of" and "as at" have different meanings - the former is a situation expected to continue for some time, the latter is a snapshot of something that may well change quickly - e.g., a financial balance, exchange rate, weather forecast (e.g., "As at 1pm on 25/2/13, we continue to expect the storm to hit London within the next 24 hours").

It's about how stable over time the assertion is expected to be. "As at" communicates the important context that the assertion is only reliable at a particular point in time.

Matt wrote: "I believe that "as of" means "from this day on" while "on this date" is expressed by "as at". I might be wrong about this as I am not a native speaker ."

dms726 wrote: "Matt, I am not sure I have ever heard "as at" used in the manner you describe."

Perhaps it's a British thing, but we distinguish between "as of" and "as at" in the way that Matt describes. Former is start of something that is on-going; latter is a snapshot in time. "I'm quitting as of Friday". "Exchange rate is 1:1.2, as at 25th July".