Time travel is theoretically impossible, but I wouldn't want to give it up as a plot gimmick. Isaac Asimov
The classic method as envisioned by HG Wells, the father of science fiction himself. This is science at it's purest.
Undoubtedly the one that most people think of is Doctor Who, but honestly, you can see it anywhere, even in the comedy Hot Tub Time Machine. For whatever reason, we have seemingly come to the conclusion that if we want to time travel, we're going to need a machine to do it. But that's not necessarily the case.
Though I'm a massive of HG Wells - who really did inspire the world of time travel, I have to give credit where credit is due. The Clock that Went Backwards was the first story to use a machine for time travel. Unfortunately for its author Edward Page Mitchell, it was published in a boys magazine so passed unnoticed by critics and history.
Also A Connecticut Yankee in King Arthur’s Court was the first movie to have time travelling in it in 1921.
A Connecticut Yankee in King Arthur’s Court The Time Machine and The Chronic Argonauts, HG Wells Doctor Who Back to the Future Meet the Robinsons Hot Tub Time Machine Twelve Monkeys The Clock that Went Backward, Edward Page Mitchell The Great Time Machine Hoax, Keith Laumer
By will alone
Who needs a time machine when you can travel through time by sheer will alone?
This is usually reserved for the superhero genre, but the Twilight Zone had a character travelling back in time out of pure nostalgia and desire to go back to his childhood.
One of my favourite examples of this method would be Somewhere in Time, where a man using self-hypnosis manages to send his body back in time in order to hit on a girl he saw in a photo. It's way more romantic than it sounds and a beautiful movie.
Somewhere in Time Time Travelers' Wife Heroes Twilight Zone, episode Walking Distance
Space, Wormholes and Quantum stuff
A bit of a catch-all for time travelling science so advanced it doesn't even need a car or telephone box. A real big ball of wibbly-wobbly, timey-wimey stuff. We know it's science and it's not a machine, but other than that we're not getting a clear idea of how it's working.
In Star Trek, time travel is easy. You take your spaceship, kick the engines into high gear, skim the surface of a sun and slingshot you way around and back into the past. You know, because of gravity.
Or, if you happen to be Superman, you can fly around the Earth really, really fast, causing the Earth to rotate the other way and reversing time.
The Terminator Donnie Darko Superman Star Trek, The Voyage Home
Sleep and Popsicles
I debated whether or not to include this one but figured people would assume I missed it. I consider it the cheat of the time travel world. You have a character born in the 1930s you want young, attractive and kicking ass in 2010? Stick him on ice. Wake him up in eighty years. Easy.
It's the rough and dirty method of time travel and frankly it the most unimpressive. But you can't deny, as a method, it's used everywhere.
You can also just fall alseep. Scary idea but before we had cryogenic freezing, writers simply had their character's falling asleep and waking up in the future.
Captain America Futurama Planet of the Apes Lost in Space Austin Powers: International Man of Mystery Rip Van Winkle, Washington Irving Golf in the Year 2000, J. McCullough
God or grand design
Twilight Zone, Ring-a-ding Girl
This category is really for instants of time travel which are unexplained. It happens because it was supposed to have happened. Because God intended it or because the universe required it. This is usually used in fiction where the science or greater reasoning is unimportant - the primary focus is what the characters do with this opportunity.
Christmas Carol is a good example. It's unlikely Charles Dickens was thinking much about the science behind his cautionary tale, but we can see the hints of a grand design. Old Ebenezer was chosen for this unearthly help and by gum he's going to receive it whether he likes it or not.
Groundhog Day Twilight Zone, episode Ring-A-Ding Girl Christmas Carol, Charles Dickens Memoirs of the Twentieth Century, Samuel Madden
Now we all know how, here's my question. Where and when and who do you want to visit?
Currently on my list is the Titanic, the Library at Alexandra, Oscar Wilde, Nikolai Tesla and Ada Lovelace.
He will be missed.