Rescue Romance

This staged rescue is a common simple plan to get a couple together and one of the ways writers get a relationship hampered. They might not have even met or talked to each other previously.

A typical set up goes something like this:
 * The main protagonist meeting a female character (Love Interests, etc) by saving them from being hit on by thugs or villains. The thugs grab the female by the hand and pull her away to have a "cup of tea" with them or become violent as she stubbornly rejects their advances (may involve knives). Enter the hero, either hitting the offenders in the arm, or entering the scene by kicking the guy (head, solar plexus, etc) and saving her from forceful romantic advances.


 * If Rescue Romance is a manly man's reward for being a valiant protector, the Florence Nightingale Effect is a womanly woman's reward for being a caring nurturer.

Movies

 * Disney/Pixar
 * Tramp saves Lady from nasty dogs and the annoying muzzle. Then she and the other dogs save him from the dog catcher.
 * Ariel saves Eric from drowning who falls in love with her voice. Then he saves her.
 * Back to back, as Belle finally gets fed up enough with the Beast that she escapes the castle... only to get attacked by wolves that the Beast saves her from. Belle drags his unconcious, wounded body back to the castle and patches him up, leading to the first conversation they have in the film that isn't openly hostile. Cue Relationship Upgrade since now the Beast is more conscious of being nice to her and Belle is finally able to let go of her conceptions of him.
 * Hercules tries this with Meg, but she's less than impressed. Then, she saves him from a falling column, taking the impact and confessing her love for him. He finally saves her from the afterlife and their romance is sealed.
 * Aladdin meets Jasmine by saving her from having her hand chopped off. He then goes on to save all of Agrabah, leading the Sultan to decide that he's worthy of marrying her.
 * 20th Century Fox/Blue Sky Studios
 * Blu saves Jewel from falling and after they kiss. As a result, Blu finally can fly because of their love.
 * Hanna-Barbera
 * Yogi Bear finds Cindy Bear at the Chizzing Bros. Circus to rescue her from that greedy men and their Dog Mugger.
 * DreamWorks
 * Princess Fiona and Shrek meet when he rescues her from a tower and a dragon. She thinks he's supposed to be her Knight in Shining Armor with whom she will share True Love's Kiss until he explains that he's actually an ogre who was just sent there to do the rescuing so Lord Farquaad wouldn't have to. Ultimately, Fiona and Shrek end up falling in love anyway.
 * Megamind protects Roxanne Ritchi from Tighten.
 * Gia saves Alex from Dubois' poison dart.


 * Ana Lewis manages to break free from the mind control and came to Duke's rescue just before the Doctor (her brother) could subject him into turning Duke into a mindless Viper, also they share a kiss.


 * Jack Dawson rescues Rose DeWitt Bukater when she attempts suicide. She falls for him, and upon telling her story 84 years later, says that he saved her in every way a person could be saved.


 * Wonder Woman rescues Steve Trevor from a plane crash, resulting in their friendship.


 * Garrett saving Kayley from Ruber's thugs


 * Kovu rescuing Kiara from the wildfire.


 * Tad Stones saving Sara Lavrof from falling down a deep pit.


 * Pocahontas saving John Smith from being executed by her father reuniting her tribe.


 * Sinbad rescuing Marina from the Roc Eagle.


 * Steven Universe defending Connie Maheswaran from Holo-Pearl.


 * Beans kisses Rango for saving her from the bandit's clutches.