Have you always had a burning desire to travel through time? Good news! You are already doing it. You are moving through time at quite the speed, even if you’re sitting comfortably at home on your sofa. There’s only one issue, you can only move in one direction.
The more pressing question is can we travel towards the past? As far as we know, no. There is no known mechanism that allows us to physically transmit matter or information from the present into the past. Time travel requires going faster than the speed of light and we haven’t found a hint that it’s possible. It is highly unlikely that traveling to any arbitrary point in space-time can actually happen.
This could be an accurate if somewhat disappointing “case closed” on the question, but there are two reasons why it is important to chat about time travel. Number one, time travel would be great and I totally want to do it. Number two, it is good to challenge one of the biggest unknowns in physics: time.
What is time? We don’t know. Why is time going in only one direction? We don’t know. Must it always move forward? Maybe, but we don’t really know. Is time continuous or quantized? Is it made by tiny but finite units? We don’t know. This is the state of our ignorance regarding time.
Relativity Gives Us (Some) Hope To Go Back To The Past
This ignorance has consequences for our theories. Once theories are formulated, people like to play with them and push them to the limit. Black holes and the Big Bang were once mere ideas out of general relativity. Could the same happen for time travel? That's unlikely but in the interest of fairness, there are special solutions that allow the creation of regions where it is possible to cheat physics and go faster than the speed of light.
You can start simple and have a solution for closed, time-like curves. You travel back in time simply because time is a loop. Think Groundhog Day but with more math. Alternatively, you can have a wormhole, a bridge between two distinct points in space and time. The only issue is that both of these require conditions that we have not witnessed in the known universe.