Planet of the Apes (1968)
Put briefly, astronaut Taylor (Charlton Heston) unknowingly goes on a journey to a future Earth where apes have guns, don't believe in the possibility of flight, and rule the planet. Humans are still around in this future, but are mute and quite unintelligent.
The best thing about this movie is that they talk about real scientific theories in explaining how they traveled through time. "According to Dr. Hasslein's theory of a vehicle traveling near the speed of light, the earth has aged nearly 700 years since we left it, while we've aged hardly at all," says Taylor in the opening scene of the movie.
Actually, it's based more on theories of a man named Albert Einstein. His special theory of relativity says that time for a moving clock slows down.
In 1971, the Hafele-Keating experiment proved this with very accurate atomic clocks. One was on Earth and another was flown around the world on a passenger jet. When the jet landed, the clock on the jet was about 50 nanoseconds behind the one on Earth, just like Einstein predicted in 1905.
"This means, for a space traveler traveling close to the speed of light, that this effect will happen dramatically. A few years will pass for those on board, but when the rocket lands, decades will have passed on Earth," explains Mallett.
So as Heston and his crew travel near the speed of light, they would in fact be traveling through time relative to those of us on Earth. Getting a ship close to the speed of light is the practical challenge in this case, but very plausible in theory. The portion of the movie in which apes control the planet is a little more questionable.