The last time that Doctor Who fans saw companions Amy Pond and Rory Williams, they were being shot back into the history of New York by a Weeping Angel (in the Universe of the Doctor, a creature which subsists by stealing time from people, sending them into the past and feeding off the potential time they had left from their original 'future' point). Because of some timey-wimey fragility introduced to the system, this puts all of New York off limits to the Doctor's future travels, meaning he can't go and 'save' them from the past. But the appearance of their names together on a Manhattan gravestone signals that they not only survived, but grew old together, and stayed in New York.

Exactly when and where and how they lived is not as clear. The gravestone, seen near the end of the episode, is scant on information, reciting only:
IN LOVING MEMORY
RORY ARTHUR WILLIAMS
AGED 82

AND HIS LOVING WIFE
AMELIA WILLIAMS
AGED 87
This tells us a few things. Firstly, contrary to the general human custom there are no dates on the gravestone. Since it is usually within the power of the dying person to dictate what information will be relayed on the grave, it must be presumed that this is intentional on the part of Rory and Amy (especially Amy, who ends up being the longer-living one in this timeline). Had they wished to convey more information, to send a signal or a code, to call for help in some way, this gravestone could have served that purpose. Rory and Amy were schoolmates, and so were about the same age (at least, before the Doctor entered their lives, and discounting the 2,000 years that Rory spent guarding the Pandorica as a plasticene Auton in the form of a Roman soldier). And so the grave marker additionally indicates that Amy spent the very last few years of her life widowed.

The gravestone is itself in good condition, not showing any ravages of time. Since the Weeping Angels had been shown previously in this episode to have been sending victims back to the year 1938 (74 years into the past from the 'present' of the episode), it is logical to presume that approximately the same transit was made for both Rory and Amy in their next sending-back. If both were nominally around 30 years old when they arrived in the past, then Rory would have lived 52 years more after their trip back in time, and Amy 57 years more, putting their actual death dates around 1990 and 1995. Interestingly, Amy and Rory each had already traveled to times and places within that decades-long range in the Doctor's company. For example, one adventure had Amy and the Doctor visiting Winston Churchill sometime late in the Second World War. And in another-- much closer to their 'new' home-- Amy and Rory joined the Doctor in visiting Richard Nixon's White House around the time of the moon landing. Simply put, if Amy and Rory had really, really wanted to see the Doctor again, they could have shown up at points such as these. But instead, it seems, they finally took the opportunity they had previously mulled, to settle into a quiet, Doctor-less life.

The only other clue as to their fate, which adds not much at all, is the last page of the novel written in the past for the Doctor to discover in the future, recounting the adventure. On the last page it is written: "By the time you read these words, Rory and I will be long gone, so know that we lived well, and were very happy." Taken at face value, this reinforces the message of the tombstone: Amy and Rory remained happily together into old age and death.

Now I'm going to go out on a limb and speculate a bit. Firstly, Amy and Rory popped back into Depression-era New York with literally nothing but the clothes on their backs. But Rory was a nurse, meaning that he had some medical training-- and medical training at the 21st century level, such that Rory in the 1930s would have had the skills necessary to practice medicine insofar as it was practiced at that time. I would therefore suppose that Rory quickly found employment in some hospital capacity, and possibly as a medical doctor. Alternatively, Rory spent centuries as a Roman soldier, and remarked that his head had been filled with Roman stuff, making him equally qualified to teach some little-known aspects of Roman history. Amy, meanwhile, had given up modeling to become a travel writer, an occupation she could have continued into the past, except that it doesn't seem likely that they did much traveling once they settled in New York (and after having seen great stretches of the galaxy and more of human history than most men). But, once a writer, always a writer, so at the least she could've made a success of herself authoring the sort of novels which were at the core of her final episode.

There must be some question as to whether Rory, being a man then in the prime of his life, would have participated in any of the goings on of World War II (which would be his second go at that war, having lived through the blitz once as a long-enduring Roman soldier). But the better part of valor for this pair would seem to be in sitting back and avoiding further fiddling with the course of human history, so I'll give them that supposition as well. And lastly I will propose this. As flexible as time and space are in this reality, it is not at all absurd to suppose that somewhere, somehow, sometime, the Doctor may yet see Amy and Rory again (if even in their middle-aged or elderly states).

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oakling responds:
What I don't understand, and which still kind of pisses me off, is why the doctor himself wouldn't be able to, say, land the TARDIS however far outside of New York as is necessary and take a fucking train in to visit them. I mean what is with "wah sob I never get to see Amy and Rory again because the universe will disintegrate if the TARDIS comes to New York EVER AGAIN"? come the fuck on, is what I am saying. It would have made more sense for them to invoke some kind of gibble-babble about how he can't cross whatever timelines by visiting people who he knows are dead. Even though that wouldn't have made sense either because he visits Shakespeare and all. Basically, none of it makes any sense and they should have let those two die a hero's death by dying when they jumped off of the building. second best, had it properly save everyone and leave them alive: none of this "oh, er, um, there must have been one weeping angel left over? but it's certainly nothing to worry about leaving HERE IN THIS GRAVEYARD" nonsense.