Rescue Introduction

A new character is met when the protagonists rescue them from a tricky situation, or when they rescue the protagonists. Often creates a strong emotional bond to new character, as said rescue adds instant dramatic tension to any first impression. Depending on attitude and reaction of character, can act as an Establishing Character Moment. Will sometimes result in a Rescue Romance. Compare to Come With Me If You Want to Live, which has a similar introduction dynamic. The rescuer can act as The Cavalry, or Big Damn Heroes depending on the situation.

While this meeting may often overlap with introducing the character to the audience as well, it does not necessarily have to- this is an in-universe trope about how the protagonist(s) meet the character.

Anime and Manga

 * In the anime Romeo X Juliet, this is how Juliet meets both Romeo and Tybalt.
 * Neon Genesis Evangelion: Misato rescues Shinji from getting squashed by the First Angel/blown up by JSDF the very first time he sees her.
 * The Breaker: Shi Woon rescues Sosul from the Torrent Clan, and while it does make a big impression on her, Shi Woon's oblivious to her feelings.
 * Nanoha rescuing Subaru in the first episode of Nanoha Striker S. Interestingly, the series ends with.
 * Barnaby from Tiger and Bunny is introduced partway through the first episode when he swoops in and rescues protagonist Kotetsu by snatching him out of a fall, Bridal Carry-style. This is Played for Laughs in that episode itself, soon becomes a Running Gag, and is even used as a Book Ends.

Comic Books

 * Spider-Man's parents were spies working for SHIELD. In the special -1 issue of Untold Tales of Spider-Man, the audience is treated to a case in which they have to save a fellow agent. When they rescue that agent, it's revealed to be  whom they were meeting for the first time.
 * In The Blue Lotus, Tintin meets Chang Chon-Chen by rescuing him from drowning in a flood.

Film- Animated

 * Hercules: Hercules meets Megara when rescuing her from a monstrous centaur.
 * Aladdin: Aladdin meets Princess Jasmine when rescuing her from a street vendor who she accidentally stole an apple from and her arm is nearly slashed off.
 * In Shrek, Shrek and Donkey meet Princess Fiona when they rescue her from her tower.
 * The Little Mermaid: Ariel meets Eric after he is accidentally separated from his ship during a hurricane, and is brought to shore to prevent him from drowning.
 * The Hunchback of Notre Dame: Phoebus first meets Esmeralda after he saves her from being arrested by his own teammates.

Film- Live Action

 * Star Wars has one of the most iconic examples of this: when Luke rescues Leia.
 * Kick Ass: Kickass is about to get carved up by Razul's gang when Hit-Girl gets her Establishing Character Moment by massacring them all. This is also the first time we see her in the Hit-Girl persona.
 * Pale Rider: The Preacher and Hull first meet when the Preacher rescues him from a bunch of thugs.
 * How Hancock meets Ray.
 * This is how Number Six introduces herself in I Am Number Four.
 * The Terminator has Kyle introduce himself to Sarah J. Connor by extricating her from Tech Noir just as the T-800 closes in for the kill.

Literature

 * The Vorkosigan Saga: How Miles meets Taura. However, he was also on a mission to kill her. It Makes Sense in Context.
 * Murtagh rescuing Eragon in The Inheritance Cycle.
 * In Much Fall of Blood Erik and Bortai first met when she was escaping from Gatu's forces, and he offers her shelter, with unintended consequences.
 * How Merlin Athrawes introduced himself to then Crown Prince Cayleb in Off Armageddon Reef.
 * Dream Park. The players first meet Alex Griffin's character when they find him sinking in quicksand. One of them teases Griffin by making him prove that he's worthy to join the party before they rescue him.
 * Neverwhere: How Richard meets Door when he takes her home after finding her lying and bleeding on a cold London Above street.

Live Action TV

 * Several times throughout Doctor Who, but explicitly in the episode "Rose", where Rose's first meeting with the Doctor is him grabbing her hand and yelling "Run!"
 * How Dov meets his girlfriend in season 2 of Rookie Blue. She's a bomb disposal expert called in to help him after he steps on bomb-trigger trap, and rescues him from said trap.
 * In Suits Harvey and Mike meet this way when Harvey and Donna conspire to rescue Mike from the police chasing him in the pilot.
 * Lost Girl: How Bo meets Dyson's New Old Flame Kiara. And how Dyson re-connects with her after several centuries of separation.

Video Games

 * Bio Ware likes this trope for party members:
 * Mass Effect: Ashley, Liara and Tali from the first game. Archangel,, arguably Jack and Grunt from the second game. In retrospect,
 * Dragon Age Origins: The Player Character for Flemeth, and then more conventionally: Sten, Shale and arguably Wynne.
 * In Dragon Age II, Hawke and family meet Aveline by saving her and her husband from darkspawn. Flemeth again introduces herself after rescuing Hawke and company from darkspawn.
 * Baldurs Gate 2: Branwen, Dynaheir, Viconia, Yeslick and Xan in the first game; Aerie, Cernd, Haer'Dalis, Viconia (again!), Mazzy and arguably Minsc and Jaheira in the sequel.
 * Knights of the Old Republic has you rescuing Bastila (or as she'll insist, her rescuing you...)
 * Metal Gear Solid 2: The president and Otacon's sister. Although I don't know if  to count for this trope.
 * In the first Metal Gear Solid Snake meets Otacon by rescuing him from the Cyborg Ninja.
 * The videogame adaptation of The Godfather has you meet Frankie when she's taken hostage by Vito's would-be assassins. The same event also leads you to become personally acquainted with Vito.
 * In Half-Life 2 you meet Alyx Vance after she rescues you.
 * In Final Fantasy VI, Celes is introduced just as Locke rescues her from the jail-cell.
 * In Mitsumete Knight, this is how the main protagonist meets Sophia (punks harrassing her), as well as one of the two ways to meet Laury (dangerous dog about to attack her). Depending on if one of them is the girl you target the ending of on your playthrough or not, it can evolve into Rescue Romance.
 * In Pop'n Twinbee, Light and Pastel first meet Madoka when she begs them to rescue her grandfather Mardock, a benevolent scientist whose personnality changed into a deranged megalomaniac world-conquerer after he hit his head on a pole. They ultimately manage to turn him back to normal, making the three youngsters the best of friends ever since, and Mardock one of Light and Pastel's most trusted allies.

Webcomics

 * In Drow Tales, while looking for Faen, Ariel discovers Freyja and Myou in a dungeon. (seen here. (Spoiler Alert!))

Web Original

 * In the Whateley Universe, Team Kimba first meet Skyhawk when he swoops in and plucks Chaka off the roof of a car of badguys just before it crashes. They first meet Anti Hero The Lamplighter when he mistakes Chaka and Dyna-Man for supervillains and tries to kill them.