The concept of time travel in itself is imposing.
Time, being considered as a dimension, cannot be directly altered by us (for now). We cannot merely move along this dimension, because we are not sure how it flows. If we are to walk along this dimension, anomalies are bound to happen.
Most of these paradoxes arise because all ways of travel violate causality. An effect always succeeds in the cause.
The grandfather paradox is one of the famous ones. You cannot only go and kill your grandfather because you are an "effect" that is directly affecting the "cause". If you, however, do manage to kill your grandfather, you would create an inconsistent causal loop. It is a type of temporal paradox.
Consider another scenario/series of events. Consider I'm a singer. To be successful, I need to compose great songs, which I cannot currently come up with. So, assuming I can travel into the future, I go and steal one of my songs which is a hit, months from now. I use the stolen song and claim it to be my composition. A month from now, I realise I have flown my song from the future. In this case, I don't know how the song was created in the first place. It is a particular case of the temporal paradox commonly referred to as the bootstrap paradox. [Seen in the movies Somewhere in time, Predestination, etc.]
Another one is the requirement of all events happening in the exact order in which they occur. Consider I travel back in time and meet my parents when they were young. If in a series of events, I was the reason why they married each other, I would create a consistent causal loop. In this case, I would be required to always go back in time to let those series of events happen so that I can sustain my existence.
The problem with all the three scenarios is that all of them violate causality laws. Now there are counter theories that have come up to explain these scenarios. One term being Retrocausality. Retrocausality allows possible events to affect occur before the cause. Then there is the theory that fluctuations can prevent a paradoxical occurrence from taking place. For example, in the grandfather paradox, you would not be able to kill your grandfather as some event, like the gun malfunctioning will prevent you from doing so. Another possible explanation is that you can only travel back in time to another universe and not your own.
The question is if time travel is possible in the first place. If it is, why isn't there proof of it from the past? Stephen Hawking experimented testing this. He invited time travellers to a party. The invitations were given after the party took place. Based on the invitation, people should've turned up from the future. But no one did.