As the other excellent answers have shown the movies are rich of various overarching themes, be they political or philosophical, about a hero or his city. But on a very personal level they are about a man, in particular Bruce Wayne's struggle to cope with the tragedy he experienced as a child and to overcome the resulting grief, fear and anger that manifested in the personality of Batman, something he does not achieve until the end of the whole trilogy. They are much less Batman movies than they are Bruce Wayne movies.

While The Dark Knight, devised rather as a sequel to Batman Begins than the 2nd act of a trilogy, might transport this general idea to a lesser degree, The Dark Knight Rises, clearly devised as the final chapter, goes out of its way to reconnect to the 1st part and its themes in order to close the overall storyline and Bruce's character arc. -- Anecdotally I've also heard people complain that for a Batman movie The Dark Knight Rises "doesn't feature enough Batman" and this is a consequence of its concentration on Bruce and his story and also ties it well to Batman Begins, whose concentration on Bruce's backstory is a natural consequence of its nature as an origin story. -- But the visual image that best establishes this connection between the first and third movie is the pit analogy.

The fear that drives Bruce's whole later life and his definition as a hero starts when he falls into the well on the grounds of Wayne Manor at the very beginning of Batman Begins. And while this fear starts out as a simple natural and unambiguous fear of beasts, through the opera, its imagery and Bruce's urge to leave it gets inseperably tied to the much deeper and defining fear he encounters at that night's tragic events and that will never let go of him from then on.

As a result of this night he is a torn man and has pretty much lost his will to go on. When he tries to revenge his parents during his college years by downright killing Chill in public, he knows that he will go to prison for this for a very long time and he doesn't care. All he has is this anger he wants to let loose. After this fails, he becomes a lost and wandering soul, until he finds a way to take his fear and his anger and channel them into something better, after he has seen that the League of Shadows and their leader Ra's al Ghul have taken the wrong path. But this way to somehow supress his fear and anger and put them to use has the price of abandoning his past self in favour of the persona, or rather symbol, he developed for this purpose. And this comes at the expense of his personal life and its relationships:

Alfred: You're getting lost inside this monster of yours...It's not just your name, sir, it's your father's name, and it's all that's left of him. Rachel: No, this is your mask. Your real face is the one that criminals now fear. The man I loved, the man who vanished, he never came back at all. But maybe he's still out there somewhere. Maybe someday when Gotham no longer needs Batman, I see him again.

Then we have The Dark Knight, where the city seems initially on a good way and things start to clear up, upto a point where Batman might not be needed anymore, since there is a true hero arising, in Harvey Dent, a man that Bruce could have been and who in fact starts to lead the life with Rachel that Bruce could have had. But things get out of hand when somone emerges that puts Bruce and his ideals to the limits, one that gives him the same perils he once went through with his parents and questions his one rule. And in fact he has to break this rule at the very end and this together with Rachel's death shatters all his hopes that Batman is only a temporary solution. And it is Rachel who identifies a key problem, namely that Batman is not only a device to help Gotham, but even more so one to help Bruce:

When I told you that if Gotham no longer needed Batman we could be together, I meant it. But now I'm sure the day won't come when you no longer need Batman.

Exactly this point is finally reached at the start of The Dark Knight Rises then. Gotham does not need Batman anymore since things cleared up and indeed Batman has vanished. But this hasn't led to Bruce now living on. No, instead he does not live at all anymore, he has gone into complete recluse for years since there is not much of Bruce left that could live on. The only way out of this that he sees is being entirely swallowed by his substitute personality and dying in this process, the man completely dissolving into the symbol.

Alfred: And that's the problem. You hung up your cape and your cowl but you didn't move on. You never went to find a life, to find someone.

Bruce: Alfred, I did find someone.

Alfred: I know, and you lost her, but that's all part of living, sir. But you're not living, you're just waiting, hoping for things to get bad again...You see only one end to your journey. Bane: You don't fear death, you welcome it.

It is only after he has been broken by Bane and left in the pit that he regains his will to live and to move on. And this pit is pretty much the well Bruce fell down when his whole misery started and that he never got out of alone. But this time it is Bruce alone who achieves to climb out of this well by himself. And this is the important difference here, since there can be no salvation for him if he doesn't strive for it. After nearly 30 years he finally has "learned to pick himself up" and carry on from his sorrows. (You could as well metaphorically relate this pit to the Lazarus pit known from the comics in the way that it healed Bruce's mind and spirit even more than it healed his body.)

In particular he finally has found the way to separate the man from the symbol and sacrifice the symbol to remain in the spirit of the city forever without also sacrificing the man. He has not only saved the city. No, the city, as nicely pointed out in another answer, will always be in danger. He has first and foremost saved himself from Batman and the feelings of fear and anger that manifested through him.

On the BluRay of The Dark Knight Rises the filmakers also talk a bit about the character of Bruce Wayne, reinforcing the above notions:

Jonathan Nolan: I think being concious of never straying too far from these being films about a man. There's a world, there's a city, there's amazing characters but it's really the story of a guy who decided to do something very unconventional, illegal, dangerous out of a somewhat broken sense of righteousness and justice. I think what's fascinating is watching this guy trying to find that path through it. Jordan Goldberg: The reason why the story is soulful and relatable is that beyond the spectacle, beyond the conventions of the genre Chris is telling the story of a man's journey to rediscover the will to live. Christian Bale: How much longer can he allow this pain that has happened to control what he does with his life? And at what point does this start to become completely self-destructive? David S. Goyer: We try to treat Batman, the Batman costume, the Batman personality almost like it were an addiction. That he's addicted to it, that he's addicted to the anger, that he's addicted to the violence, that he's addicted to the suit. And that's all he really lives for...The lesson that Bruce has to learn is, at a certain point he's gotta walk away from Gotham and he's gotta walk away from his anger...He's depressed. So the only way to escape that, the only way he's gonna live and not die, the only way that Bruce can really free himself of these demons is if he lets go of the anger. And in order to let go of the anger, that means letting go of Gotham, letting go of Bruce Wayne, letting go of all of it. Christopher Nolan: Jonah's draft of the script always had this great idea of this underground prison that you have to climb out of to escape. I thought it was the perfect metaphor, just really in tune with all the thematic elements of the film. David S. Goyer: We wanted to bring back that imagery of when he fell into the well, his dad brought him out of the pit and now it's time to bring himself out of the pit.

So on the bottom line this trilogy is the story of a man (or a boy) trying to overcome the death of his parents and the resulting fear and anger, and thus growing up in the process.

In the 1st act he finds a way to channel this fear and anger into a symbol, but also one that starts to consume his normal personality. In the 2nd act he loses all ties to this ordinary personality and all hopes for a normal life. And in the 3rd act he finally overcomes this artificial personality and the fear and anger manifesting in it by separating himself from the symbol and moving on with his life.

This is of course not to say that this is the one and only overarching theme of the movie, but it is an important one and the one representing the personal/character viewpoint of the whole story.