Also not my field, but I'm got some experience with very selective admission committees and wanted to add an important point to all the great ones mentioned above:
Don't take it personally
Seriously. You would not believe how, at a certain point, this process is largely arbitrary. Because, really, it's not like the departments are looking carefully at each application and saying "this person meets our standards, this person doesn't" and all the former get in. No. They do that, then look at the still large pile of people who meet their standards and they'd like to accept, and then somehow figure a way to whittle it down to the number of slots they actually have available.
It bothers me when I see people on results page say "oh, I knew my GPA [or GRE] wasn't good enough" or "I bet it was because I didn't have any publications." If your GRE/GPA is way below the average for your program, that might be the case. But for people at or above the average, it is literally impossible to guess why you weren't accepted and someone else was. (It's not impossible to know, you could ask them and they might tell you. But it's impossible to guess.) Once you meet a certain standard on the basics and you're on the shortlist, you can bet that what gets you accepted or rejected after that point is entirely out of your control. Departmental politics, a particular faculty member's ability to take on another student, the profiles of the students accepted last year, the profiles of the other students who will probably be accepted this year, unconscious biases (or affinities), funding issues ... Or a billion other things that could affect the decision — all of which are out of your hands, and none of which are even really about you.
So if you get rejected, even if you get rejected by all of them, don't take it personally. Take a good look and if you have obvious deficiencies make a plan to correct them, but if you don't, don't drive yourself crazy trying to find what small flaw caused them to reject you. Because, a lot of times, it's not you; it's them.
