One major problem for time-travel is the apparently unavoidable paradox: if the past is changed it may affect the fact that time travel happened in the "first" place. Never mind whether it actually happens - the problem is that time-travel allows it to happen.
The paradox can occur in many variants, but its logic is essentially always the same. Astronomy trek lists 5 possible paradoxes, but I would argue these can rather be understood as 3 possible ways to avoid the one and only paradox: What are called there "Predestination or Bootstrap paradoxes" are both basically time loops in which the information/object/person travelling to the past is part of the one and only version of reality which is self consistent and unalterable - what I call below "Static History". Alternatively one can imagine that time travelers can actually change things with respect to a pre-existing "unchaged" version of reality ("Grandfather" and "Let's kill Hitler" paradoxes). In this case we need certain changes to be "magically" stopped from happening (Timeline protection hypothesis, and similar ideas), or alternate timelines /universes are invoked, which correspond to the "Dynamic History" and "Multiple Histories" scenarios below. Note I have said "avoid", as these are not solutions - a real solution would solve the problem altogether and pave the way to real time-travel!
SCENARIO 1: "STATIC HISTORY"
All jumps back in time are part of a history that is fixed, "already written". A God-like being could see it all before themselves, with all interferences from the future on the past already "built-in" in the one and only timeline that ever exists.
It is important to note that the situation described above is an "open" time loop. Here an "open" time loop happens ONLY ONCE, the only difference with "ordinary" time flow being that there is an effect that precedes its cause. If we follow history in such a situation, at one point we'll see something from the future having an influence on present events. Further down the line, when the point comes at which the trip to the past departs, this is not happening "again", it is the SAME and ONLY one trip to the past that took the object/person/information to an earlier time.
If the interference happens far enough in space or time from the time-travel event, it would seen that we can think of it as being "diluted" in history and not having any bearing in directly causing the travel back in time. But I would argue that any time travel to the past incurs in the bootstrap paradox, because we really cannot conceive any alternate history where this has not happened, so we never really can be sure that an interference was "diluted" and had no bearing in the events that led to time-travel.
This also implies that a "predestination paradox" is created, because it's the travel back in time what caused time travel to be invented in the first place, or, as said, at least we can't rule it out.
In any case it is also important to note that in this scenario we need to accept that freedom of will is just a fiction, that in reality everything is already set to happen the way it does.
It is also important to point out that the single history incorporates all actual jumps in time, not all potential ones. The world view where all potential events do actually happen by calling a parallel universe into existence where they do, can still be added on. In fact it would, in a way, restore freedom of will to this scenario.
SCENARIO 2: "DYNAMIC HISTORY"
In such a scenario we can imagine, as many movies have, that time travelers to the past try to change things. Each time time do, history is overwritten. This means we can never be sure that any of these changes prevents time-travel from being invented. This is basically the "grandfather paradox". A typical solution is that such changes are "magically" rendered impossible. So we preserve the "predestination" of time travel being invented, but it seems a rather "ad hoc" and unscientific solution. But if changes are allowed the problem (grandfather paradox) is that these changes always could change history in a way to prevent time-travel from being invented. And if they could happen, we must address this possibility in order to sort out the question of whether time travel is possible in the first place.
Clarification: in movies we often see a "local" version of the paradox in which the trip back in time changes things in a way that the reasons for traveling back disappear, such as fixing a problem or killing your grandfather. This situation can then be "fixed" in turn to make things happen again with more and more time travel events (and here the movies usually stop), but the overall outcome of all attempts is that history is indeed changed, and these changes could, as said, prevent time-travel from being invented. In other words, we can always "fix" smaller paradoxes but the ultimate paradox of time-travel being prevented by a changed history remains and can never be ruled out.
The curious thing about this paradox is that one possible answer is "so what?" If history is rewritten without time-travel being invented, so be it. That one version of history is overwritten by another one does not mean that the older one never happened (this would be scenario 1), so the fact that time-travel is never invented in the new version does no really clash with the fact that someone/something traveled back "before" history is overwritten. Basically this means that that grandfather paradox does not really exist!
Another corollary of this scenario is that (open) time-travel loops will keep happening until in one of the iterations time-travel is actually prevented from being invented. This means that time-travel loops will keep happening until a version of history is generated where it is never invented. This in turn would mean that time-travel is doomed as, every time it is invented, the universe "naturally selects" in favour of histories where it is never invented, as these are the only ones to continue unimpeded.
SCENARIO 3: MULTIPLE HISTORIES