My Anglophile girlfriend and I have been watching the reboot of Doctor Who--the iconic British tv show about the quirky alien time traveler with the eight foot long scarf¹ and the celery in his lapel².
¹ not any more
² Really! But again, not anymore.
I have always loved a well-crafted time travel story--one that tries to make it as realistic as possible rather than just sightseeing through the past or the future. Of course, "as realistic as possible" has kind of a squishy meaning when you're talking about traveling in time with an alien flying a magical police call box, but for me that means that my time travel story should confront something of the potential ramifications of the time travel.
Doctor Who plays fast and loose with time on a case by case basis. The Doctor is not allowed to go in and change the past, except for every episode where he goes in and saves dozens or hundreds of people--entire planets some times. He can't effect history, except when he can. The show is fully aware of the absurdity of this--what the Doctor can and can't do is limited by the dramatic necessity of the story: in a recent episode, the Doctor's Companion. Amy states it thus, "So is this how it works Doctor? You never interfere with the affairs of other people or planets, unless there's children crying?"
But my favorite type of time travel story comes from those stories where the act of traveling back into the past winds up being the thing that makes the future possible. It's a narrative technique that views time as something that exists in its entirety--a closed loop--where everything that exists or will exist already exists, even if it just hasn't happened yet.
The classic examples of this trope are in Slaughterhouse-Five and The Terminator. Some of the best episodes of Doctor Who have indulged in it as well. Done incorrectly, it can feel lazy and a cheat--Bill and Ted's Excellent Adventure hung a lampshade on this when the heroes, needing to free a bunch of people from jail, realize they can go back in time later to plant the keys they need right where they need them. In a comedy like that it's fun, but in Doctor Who--a show which has an exceptionally high body count³ for what is ostensibly a show for children--it can feel like a cheat, or, alternately, you'll start to wonder why the Doctor doesn't just go back in time to save the lives of everyone who he's ever cared about who died, beyond just narrative necessity.
³ A first season episode has the Doctor weeping with astonished joy and shouting, "Everybody lives, Rose! Just this once, everybody lives!"
Like most things involving time travel, you can very quickly get bogged down in the paradoxes. The scientific and metaphysical problems of causality (that is, cause preceding effect) have been more or less a recognized central pillar of existence since Aristotle got to cogitatin'--it makes intuitive sense, and its been accumulating more and more math behind it. And if there's math involved, it must be true.
So you get things like the Novikov self-consistency principle, which states that contradictory loops that would create time paradoxes are impossible to create--the universe is self-regulating in some capacity, like a natural enforcement by Time Cops.
Science fiction doesn't have to follow the math or limit itself to what is actually possible. When the Legend becomes math, print the Legend.
Most time travel stories suffer if you think too hard about them from the Predestination and the Ontological Paradox. They both concern the chance with extensive time travel to concern elements that have no clear creator--the Ontological paradox focusing on physical objects, while the Predestination paradox focuses on ideas. For instance, Doctor Who has a character who, at some point in the future, will be given a screwdriver by the Doctor (it's a very technologically advanced screwdriver. It's sonic!). In her first appearance, she travels back in time, and winds up giving the screwdriver back to the Doctor.
Simple enough, but it means that the screwdriver has no creator--in fact, never was created, and will never be destroyed. At the earliest point in time, it is given to the Doctor, travels with him, and is given by the Doctor as a gift to a woman who will eventually travel back in time and give it to the Doctor. It exists in an eternal loop. Which raises the question of, well, what happens to it when it gets damaged? It exists in a loop, so when it picks up the normal wear and tear that a sonic screwdriver carried by a particularly adventurous alien would experience, what happens to that wear and tear? If he drops it and it gets a dent, doesn't that mean it has had that dent for its entire existence? The fact that it has always existed, but only for a portion of time, without ever creation or destruction...hopefully you're starting to get why this is problematic paradox.
The predestination paradox is a little easier, but no less problematic--it concerns an idea with no origin, transmitted back in time. If someone from the future takes, let's say, a working perpetual motion device and brings it back to me, and I then reverse engineer the device, patent it, and spread it across the planet, and my "invented" device then becomes the device that will eventually be brought back in time to me, that's just as problematic, even if it's the idea rather than the object that is without beginning or end. It's a creation without a creator! Where did it come from?
The idea that information is just as impossible to create out of nothing as matter or energy is a much misunderstood fact--I say this as someone who has spent a large portion of my life not understanding it--at its worst, it is misused by Creationists who try to claim that it means that the universe is incapable of existing without God guiding and creating everything (forgetting of course that this formulation means God is even less probable). Regardless, I think it's pretty clear to see the paradox. Under that set-up, there would no moment where such an idea would come into existence. Like the screwdriver, it would be just placed into the universe by dint of it having eventually existed. There may be some question of whether the universe or God plays dice with us, but we can probably hope that She not a fan of Three Card Monte.
As a young writer, I had plans to write a story about a man who uses time travel to try to become a remarkably prescient science fiction writer--wanting to become the next Jules Verne, or the old Jules Verne, he attempts to abuse the ontological paradox by publishing science fiction stories in the late 1890s that are remarkably accurate descriptions of the way we live today. He's hoping to become the next Nostradamus and H.G. Welles and Jules Verne all rolled up in one. Unfortunately, his writing is so terrible that no one reads his stuff, and he remains unknown. You can try to change the past, but you can't change your writing ability. And the ontological paradox gets his revenge--his remarkably prescient work has always existed, and always been ignored for being so terrible.