As far as most rational people are concerned, time travel is merely one of the more interesting topics brought about by science fiction. Like most science fiction, while time travel can seem rooted in some arcane secret of theoretical physics it eventually boils down to pseudoscience and what-ifs. However, immediately disregarding a topic as nonsense wouldn't be in the spirit of science - any good scientist evaluates all possible hypotheses rationally before coming to a conclusion. As such, let's begin to talk about some of the "possible" ways that time travel has been theorized.
One common element of all time travel theories is the mechanic of "time dilation". According to Einstein's theories of relativity, as an object's velocity approaches the speed of light, that object's perception of time slows down. An example of this would be if you were an astronaut flying in a spacship that could fly at a velocity near the speed of light. Once you had returned to Earth from your voyage across space flying at ludicrous speeds, your perception of time would slow down and you could feel as though hardly a day had passed and you would barely age. Meanwhile, everyone back on Earth would have experienced time at the normal rate and as such would have died of old age centuries ago by the time you return. This would not be a mind trick or losing the track of time - your being would physically experience time at a different rate depending on your velocity.
Time dilation alone technically not time travel, however. Although it allows time to be "stretched" or "squeezed", you still cannot skip a moment or jump backwards in time. However, this may not be the case if we were to introduce another object called a "wormhole". A wormhole is essentially a tunnel through space with an entrance and an exit some arbitrary distance apart. As an object enters one end of the wormhole it comes out the other end instantaneously, regardless of the distance between the entrance and exit - allowing for faster-than-light travel.
Now, let's suppose that we had a wormhole with one end at point A and another end at point B. If we were able to accelerate the exit of the wormhole near the speed of light and then bring it back to point B, the exit would experience time dilation and appear "younger" than the corresponding entrance at point A. This means that if a person were to enter the "younger" end at point B and exit at the "older" end at point A, they would appear to emerge the older end at a time that the older end appeared the same age, which would appear to an outside observer as traveling back in time.
The above situation is a fun thought experiment, but there are some obvious flaws. Firstly, it would require more energy than has ever been produced to accelerate an object to a velocity that is arbitrarily close enough to the speed of light to experience siginificant time dilation. Secondly, while wormholes are compatible with general relativity, they are yet to be detected or observed and as such their existence has not been confirmed.
Another object that theoretically allows for time travel by considering the effects of time dilation is a proposed phenomenon called "cosmic strings". Cosmic strings are one-dimensional objects of incredible lengths supposed to have been formed during the creation of the universe. To grossly oversimplify, cosmic strings are like cracks within the fabric of spacetime that cause gravitational distortion in the space surrounding them. Another important feature of cosmic strings is that traveling a full circle around the string would take less than 360 degrees due to the gravitaional distortion they cause. With the help of time dilation, an object traveling at immense speed passing by the intersection of two cosmic strings could create what is known as a "closed timelike loop" and travel backwards in time.
Much like supposed time travel with wormholes, cosmic strings' existence has not yet been confirmed. Also, the efficiency of such a method would be terrible in terms of energy usage. According to Richard Gott, the Physicist who introduced the idea of cosmic strings, to travel back in time just one year would take half the total energy of an entire galaxy. Also, because this method would rely on some sort of object traveling through the intersection of cosmic strings, one would not be able to travel back further in time than the creation of the object.
Although it may be fun to consider the theory that allows time travel to happen, let's consider for a moment the theory that disallows time travel from happening. The interesting part of this theory is that it isn't strictly physics - some of what will be mentioned is more philosophy than physics.
Imagine you had a fully functioning time machine, and decided to travel back in time to meet your grandparents. By going back in time, you will be altering the timeline in which you were born in. This in and of itself does not disprove anything until we take into consideration something known as the Butterfly Effect. The Butterfly Effect is a principle of chaos theory that boils down to the fact that any action (no matter how insignificant) has consequences, and these consequences have the possibility of snowballing into something incredibly significant. If you were to travel back in time and influence the timeline in which you were born in any small way, the Butterfly Effect could cause a scenario in which your grandparents to not meet each other (or any other general scenario in which you would never have been born), therefore creating a paradox. Although you have traveled back in time to meet your grandparents, in the timeline that you have created you will have never been born and therefore would never have existed and been able to travel back in time in the first place. This thought experiment in which a time traveler would create s state in which time travel is impossible by traveling back in time is known as the Grandfather Paradox and is a major argument in disproving the reality of time travel.
To find another huge flaw in time travel, we need to look no further than our old friend Isaac Newton and his fundamental laws of physics. Say you did have a time machine, and were able to go back in time to meet your grandparents and not cause some horrific accident that erased you from existence. You would still be breaking a fundamental law of physics - matter cannot be created or destroyed. Simply by pushing the button on your time machine and vanishing into thin air, you would be effectively subtracting matter from the present and adding it to the past. Another scenario in which this rule-breaking is demonstrated is the science fiction trope of a time traveler gifting their past self some sort of trinket. The time traveler would come into possession of this trinket by being gifted it, then passes the trinket along to their past self only for their past self to keep it until gifting it to their past self, creating a time loop in which this trinket is never actually created - only infinitely passed through time.
These scenarios are all failing to mention the fact that there have been no time travelers coming to visit from the future so far. In 2009, the late physicist Stephen Hawking threw a party open to any time travelers, but did not announce it until after the party had ended. As expected, no one showed up to the party. This shows that time travel will never be possible - if it were, then one person would have heard about the party, invented the time machine, and then traveled back in time to attend the party (assuming they had nothing better to do with the technology). All of the above arguments were also considered by Stephen Hawking, and as such he proposed the "chronology protection conjecture". This conjecture essentially states that the fundamental laws of nature prevent time travel on all but a submicroscopic scale.
As demonstrated, time travel is only possible under optimal, edge-of-science scenarios and can be easily written off with a little logical thinking. This is not to say that we are correct in completely disproving such a phenomena - science has proven itself wrong many times before, and to think that we are immune to the naivety that past scientists held is nothing less than arrogant. Physics is a wide and ever-expanding field, with unsolved mysteries taking us on paths to every secret of the universe. Who's to say that time itself can't be one of those paths?