Can we travel back in time? How can we overcome the time travel paradox?

Neither the question nor the answer are simple, as they involve a lot of theories that have (almost) never been put into practice by a human. I’ll explain the ”almost” part in this article.
Physically it is impossible (or extremely difficult) to travel back in time, because nothing can exceed the speed of light (299,792 km/sec). According to Einstein, an object traveling at the speed of light would have infinite mass and a length equal to 0, a physical paradox that might be impossible to achieve.
We can however slow down time (theoretically) by doing two different things. One of them is gaining a speed close to the speed of light. If one person could achieve something like this, he would age much much slower than, let’s say, his family that remains on Earth. 
Now for the second part: a person who is living on Earth ages at a different pace than a theoretical person who is living on Mars (adapted from Einstein’s theory of general relativity). This is because Earth’s gravity is much stronger than Mars’s. Time on Mars runs faster because of the planet’s weaker gravitational field (time bent by gravity).
Now let’s unite these two!
A person traveling at the speed of light (impossible, but let’s say it could happen) would become of infinite mass. We know that any object in space has a gravitational field, it doesn’t matter how small or large the object is. An object with infinite mass would have a gravitational field so large, that time would completely stop. 

So we see that it is impossible for us to even stop time according to the laws of physics, though there are various theories that suggest we might be able to travel back in time. One of the possibilities is through a wormhole between two points in space. A wormhole is theoretically doable, but it would collapse to fast for a person to be able to use it. 
At last, the ”almost” part I was telling you about at the beginning of my post. GPS satellites orbiting Earth are known to gain around 40 microseconds a day, therefore they have to be permanently adjusted. This happens because of the gravitational pull of Earth and the enhanced speeds of these satellites in relation to a person living on the planet. 
We cannot travel in time the way we imagine in sci-fi movies, but we witness time travel everyday of our lives! When we look at the sky at nigh we see stars the way they were years, or even thousands of years in the past. Whenever we look at our Sun we see its image from 8 minutes ago. We are surrounded by past images of astronomical bodies, but we remain mere witnesses without the possibility to change the image of the past. – Roman Alexander 
Photo: Salvador Dali’s The Persistence of Memory (1931)

(The question was originally asked by Navindra Roopnarine from Trinidad and Tobago)