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troperville

The Third Man.
The Fugitive.
- Parodied in Wrongfully Accused
Smokey and the Bandit.
Race for the Yankee Zephyr (1981)
The movie The Chase. Which makes it Exactly What It Says on the Tin.
Apocalypto
The Warriors (Stern Chase example)
Raising Arizona has a chase scene that starts in a store, on foot, moves outside, goes though an unsuspecting bystander's house, outside again, and finally in a car. Also, there are dogs involved.
The first Terminator movie has a spectacular chase with both the victim and the Terminator breaking both of their legs and crawling the last stretch of it.

Honored Enemy of The Riftwar Cycle in which soldiers on the opposing sides of a war band together to escape the dark elves chasing after them to kill them for a variety of reasons.
In Robert E. Howard's Conan the Barbarian story "Black Colossus", after his defeat, Natohk abducts Yasmela. Conan chases after them.
- In "The Gods Of The North", an obviously magical woman lures Conan into a chase that brings him to her giant brothers. When he kills them, he chases her again — this time, she is seriously frightened.
In Poul Anderson's A Midsummer Tempest, chasing the escaped Rupert is half the book.
The Time Scout series is full of these, starting with the second book. Mostly they involve Skeeter Jackson.

The William Hartnell Doctor Who episode "The Chase," in which the Daleks first develop time travel and chase him and his companions all through time and space.

Dick Tracy had this as a major element in his stories as the police detective relentlessly chases a particular villain who gets more and more desperate as he realizes to his terror that this cop will not stop.

Snuggleninja is always chasing the butterfly.
In Sinfest, after Death realizes the zombie's a fugitive, the chase is on.
- Also, annually, Death has to chase down the year, who always tries to make a break for it.

Looney Tunes has many examples.
- Pepe Le Pew and Penelope Pussycat.
- Wile E Coyote And The Roadrunner.
- Bugs Bunny and Elmer Fudd/ Yosemite Sam/ various unnamed characters.
- Sylvester The Cat And Tweety Bird.
- Speedy Gonzales and various unnamed characters.
Tom And Jerry.
Dick Dastardly and Yankee Doodle Pigeon.
Gargamel and the Smurfs.
- ... etc., etc., etc., etc., we get it, cartoons involve chasing.
The entire plot of the Danny Phantom episode "Reality Trip". The Guys in White constantly pursue Danny and friends after Danny accidentally exposed his half ghost power to the public.
This troper can't recollect a full episode devoted to this, but is willing to bet that The Simpsons have done it.
The episode "The Chase" of Avatar The Last Airbender stands out among numerous episodes dealing with the Fire Nation's pursuit of the heroes.
Most of the Wallace and Gromit features end with a particularly spectacular chase. "The Wrong Trousers" is notable for having a chase sequence which occurs entirely indoors.