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My letters of recommendation have not been received by LSAC yet, should I wait until they arrive before submitting my application or is it okay to go ahead and submit the rest of the application while waiting for the letters to arrive? Thanks.

I can't speak to sending the application w/o the letters because all mine were in by the time that I applied, but I do know that those letters can take quite a while (mine took almost two weeks once the author actually sent it). It seems though, that it might not matter anyway because your application won't go complete until they get the letters anyway. If you're really itching to get your name in the hat, then go for it I suppose, but it might not make a huge difference.

If you do this (submit app. w/o LOR's), what happens if the schools get your report before the LOR's are in?

Do you have to pay another $12 for them to re-request your report? Or do they just get it when the letter comes in?
Also, should I notify the schools of this?

This is very typical. Submit the application. The schools will wait for the LORs to arrive before considering your application complete and reviewing it.

I did this because one of my letters was slow in arriving. Some schools (e.g. Harvard) didn't request the report until both letters came in, other schools (e.g. Columbia) ordered the report and then the LSAC sent them a (free) update when the 2nd letter came through the system. I haven't contacted or been contacted by any of these schools, but my status checkers all seem to be on-track so I presume that it's fine.

If the letter is only a couple of days away, it'll probably be quicker just to wait and apply as soon as it arrives but if it's more than a week or so, I'd just send in the application.

And as an aside, why the hell is the LSAC so slow in processing these reports? It's not like they're entering all the info by hand, are they? It's so byzantine and inefficient.

Thanks for the advice and suggestions, much appreciated. Going to go ahead and submit the apps, one letter is in, waiting on the others. Thanks again!