With the conclusion of Nolan's Dark Knight movies, we are now able to look at all three movies as a single entity, broken up into three parts (acts, if you will). Is there a specific theme, or set of themes, that span the entire trilogy?
Chaos and Anarchy are the greatest overarching themes of the entire trilogy.
In Batman Begins, we are presented a Gotham that is seemingly serene, as we witness a young Bruce playing in his garden, blissful and happy. Until one fateful night, his parents are taken from him in a drastic event that spirals his world completely out of control, into chaos. We are then presented the true chaotic nature of this Gotham city. Ruled by the unjust, criminal, and greedy, being sent into a downward spiral into total anarchy.
Even when Bruce returns, he is faced with a city that has only descended further into these two states. Surrounded only by criminals, anarchists, and agents of greed, he dons his mask and starts to clean up the city to the former glory we see at the beginning of the movie. After all is said and done during these events, we get a small glimmer of hope that the city can be restored after his triumph over the League of Shadows, only to be shown the hint of the true Agent of Chaos...
Enter The Dark Knight. Again, we are presented a Gotham that has recovered and is limping on from its chaotic period, only to be driven mad by the true chaotic force, the Joker himself. The Joker tests the city by bringing in his own reign of chaos, pitting ordinary citizens with disruptive and morally grey choices. Only to incite Anarchy at the end by trapping the city within itself, a repeat of past events from the League of Shadows.
The Joker then turns Gotham's greatest hope onto his side. By decimating Dent's sanity, inciting him to introduce his own version chaos, using the excuse of fairness and chance above all else, ignoring the proper system, delving into a false Anarchy, lead by a coin and anger.
This movie presents the ending the same, just when the audience think the threat is over with the apprehension of the Joker, he reveals his "Ace in the Hole". A just man turned to chaos to overthrow Batman's plans. While we see the cover up as successful, it is not without it's own consequences....
Finally, we wrap up with The Dark Knight Rises. We enter our heroes, again in a serene state, albeit decaying. Hints of uprising and plotting drive the Batman out of his vacation and hiding into what is truly going on. A plot to incite anarchy throughout the whole City of Gotham.
Bane's breaking of Batman only serve as a foreshadowing of what his true plan is; breaking Gotham. By using the bomb, Bane turns Gotham into a City-State, ruled only by the people as they see fit. Money doesn't matter, order doesn't matter, only the majority matters, and the majority wants anarchy.
Whether either theory is correct about Bruce Wayne's survival of these events. One thing remains clear; his acknowledgement that there will always be a need for a hero against the chaos and the anarchy. That there will always be a need for a silent protector, a Dark Knight. Hence his appointment of Robin John Blake as his successor, to ensure, alive or dead, Gotham has hope against any who decides to continue to place the city into chaos or anarchy.
I think the movies are about "Controlling the Fate of Gotam." First movies, I CAN control the fate of Gotham, Second Movie, I CAN'T control the fate of Gotham. Third Movie, WE CAN control the fate of Gotham.
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:
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:
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.
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.
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, reenforcing the above notions:
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.
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.
The message is straight forward and twofold: No matter how bad things are, a single person or single idea can make a difference.
This contrasts with most heroes (e.g. Superman) which follows a basic religious story (omnipotent being arrives and looks over lowly mankind, protecting us from ourselves).
In Batman the symbol is the thing, and it is bigger than one person.