Stalker Sequence

When a character has a rather vicious alien creature stealthily nibbling his or her ankles when all of said character's marine teammates have been killed off, the suspect of the week decides to pursue the genius police investigator when he realizes that said investigator knows too much or when a steel killing machine from the future thinks the character is somehow connected to a certain rebel leader and begins to relentlessly hunt said character without halt or remorse, you can be sure the character has ended up in the Stalker Sequence.

Generally, this is any scene where a lonely hero or important character is chased by the villain, though inversions can be found in Mook Horror Shows. It is a stock trope of the horror genre and has been since the very beginning, used most extensively throughout the 1980s horror wave. Interestingly, the Stalker Sequence has lately become quite prevalent in thrillers and crime shows such as CSI, inducing moments of Fridge Logic. It is used primarily to add suspense, and can be furnished with various visual subtropes, of which Epileptic Flashing Lights is most common in sci-fi. In videogames, this can lead to an Escape Sequence.

Examples of Stalker Sequence include:

Film

Seen in the Alien franchise as the aliens in question stalk the human prey.
The Predator franchise is similar to the Alien franchise, only with far more intelligent aliens, though their hunt has a code of honor to it and they'll concede victory to Worthy Opponents who survive the stalking.
The Hannibal Lecter movies have at least one of these sequences per film, oddly not usually done by Lecter himself.
The Terminator series is a bunch of movies revolving around this trope by the titular machines versus the protagonists.
The Scary Movie parodies tend to spork this trope. For example, a Killer Doll in one of them tries to pull this on a victim, only to seriously wind up regretting it as they are relentlessly pursued themselves.

Video Games

The Resident Evil franchise features lots of these sequences, with several games even having prominent aspects of the gameplay devoted to outsmarting the masters of this trope, like Mr. X and Nemesis.
Prince of Persia: Warrior Within has several parts where the Prince is chased by the unstoppable Dahaka and can only Run or Die.