• This is a great question!
The truth is that we are all travelling through time as we speak. We are all moving towards the future. But I reckon you want to know when it will be possible to travel through time “freely”, as we do in space. The answer to that involves a very good understanding of how gravity works (you can have a look at the second and third parts of my very looooong answer about it http://ias.im/47.62 ). I will explain what gravity has to do with time travel.
Gravity is all about spacetime and matter and how the interact with each other. This was Einstein’s amazing view of it. In this picture, spacetime is “kind of” a 4-D fabric and gravity is the way it bends and twists around. These bends and twists are produced by matter and matter moves through them! It is very dynamical. The main point is that, in this view, the “right” kind of matter would be able to make a bend in this fabric that could send stuff…say backwards in time. The real picture is a bit more complicated than that and there are other issues one should care about.
The picture of the apple in my profile is inside a “warpdrive”, a spacetime bubble which can move faster than the speed of light, and incidentally may be able to “travel through time”. That is just a possibility, it is far from being a reality and there are a lot of things to learn before telling when it will happen…if at all.
There are many views on the subject as well, so other scientists may not like this answer. They could argue and give very serious reasons of why this might be wrong. The truth is that it has not been established yet and a lot of discussion is needed. This is how science is done!
@iamjesus You could get on an aeroplane in New Zealand and fly to Hawaii, and still land before you started (because NZ is almost 1 day ahead of Hawaii in terms of global time zones). Or what about an astronaut in space? Their spacecraft orbits the Earth pretty quickly, so does that mean they travel through time?
I know, I know, this isn’t what you were thinking of when you asked the question. @Cesar is the man with your answer.