Artistic Title

A Title Sequence using original footage not directly from the series, but composed of graphics designed to give a sense of the nature of the show (or just to look pretty).

Sometimes this will take the form of a cartoon version of the characters (The Many Loves of Dobie Gillis, Bewitched), a flyby of the setting (Stargate Atlantis, Newhart) or simply a bunch of special effects.

This is the most common form of opening sequence in American dramatic programs and all British series.

These elements are often used in tandem with a Title Montage.

Anime & Manga

 * Elfen Lied, with its Gustav Klimt-esque paintings.
 * Blood+, though only with Colors of the Heart (the third opening sequence).
 * Averted hard in Red Garden. The opening is made of stylized silhouettes of the girls and the city, giving a light cosmopolitan feel akin to Sex and the City. The show has a different mood though.

Film

 * The breathtaking title sequence from Watchmen which combines the most iconic images of the 2nd half of 20th century ( the Hiroshima bomb, the VJ Day Kiss, Son of Sam, the JFK assasination, Vietnamese self immolation, "Flower Power", the moon landing) with the alt-U images of the Watchmen timeline.
 * The James Bond films always include trippy title montages featuring silhouettes of naked women dancing in thematically appropriate environments (most directed by Maurice Binder). The exceptions are Dr. No (the first film, which uses stylized geometric animation) and Casino Royale, which replaced the babes with surreal sequences of Bond beating the crap out of a brigade of Mooks who burst into playing card symbols as they expired.
 * The other Casino Royale 1967 had animated titles that could be best called psychedelic medieval illuminations.
 * The Pink Panther
 * The First Wives Club uses a series of 60's-style images of women, along with a song about being the perfect wife.
 * In the first version of Death at a Funeral, they use a title/credit sequence with a map where you're watching the hearse drop the coffin off where it is to be buried as the credits play out.
 * Saul Bass was the king of this trope.
 * El Dorado (1966) has the titles underlaid by a sequence of paintings by Olaf Wieghorst of scenes of cowboys at work, which do not illustrate the story.
 * The original 1978 Superman film had a terrific title sequence of what was implied to be the baby Kal-El's journey across entire galaxies before reaching Earth. The titles of the following films were considerably less interesting. Superman Returns in 2006, however, would later attempt to visualize its own version.
 * The Spider-Man sequels feature montages of still paintings that approximate the story thus far from previous entries in the series.
 * Some of the films from Mystery Science Theater 3000 episodes features this, but many more avert it in strange ways; Cave Dwellers, Pod People, Space Travelers and others were films re-edited and redistributed by Film Ventures International, strangely presented with opening and closing credit sequences from other, unrelated films. They're presumably intended to convey the feeling of the movies, but manage to fail spectacularly in some instances.
 * The new American The Girl With the Dragon Tattoo has this. It's dark and beautiful; it feels like the beginning of a James Bond title, if done by Trent Reznor -- which fittingly enough, it is.

Live Action TV

 * Barnaby Jones
 * BBC News 24
 * In its first season, The Bill used an iconic sequence showing the feet of two uniform police officers walking slowly toward camera while on their daily beat. This was reversed in the closing titles, where the officers' feet were seen slowly walking away from camera while the credits rolled. The sequence set out to show that the series was more interested in exploring the more sedate areas of daily police work rather than using a typical action sequence.
 * Cannon
 * Carnivale
 * Cheers
 * Chef!
 * Community
 * Desperate Housewives also uses cut-outs, but hasn't varied them from season to season.
 * Doctor Who
 * Ed begins with Ed driving to and through the town of Stuckyville (actually Westfield NJ)
 * Family Ties (in its first season)
 * Game of Thrones
 * Growing Pains
 * Home Improvement in its final years.
 * House uses images from the classic medical text "Gray's Anatomy." Grey's Anatomy, on the other hand, does not.
 * In Living Color took this idea rather literally. The first few seasons' title sequences had the actors painting walls, including the fourth one. The last few took place in an animated art gallery.
 * All shows in the Law and Order Series Franchise
 * Monty Python's Flying Circus used different cut-out animations for each of its four seasons; all included the famous descending foot.
 * The Nanny
 * Nip Tuck
 * Rescue Me
 * Star Trek
 * Played with on an episode of Friends titled The One That Could Have Been. Taking place in an Alternate Universe, the scene's title sequence is shot, not using the cast members as they usually are, but as they are for that one episode.
 * Season 3 of Veronica Mars used this, as the show had moved away from the High School setting of the first two seasons and the series was being promoted less as a teen soap and more as a neo-noir. Ergo, the notebook motif of the original credit sequence had to go. The music was changed also: it's still the same song by Dandy Warhols but a more brooding version is used.
 * All Ultra Series shows up to and including Ultraman Ace had silhouettes of the various characters and vehicles being shown on a background of colored stuff(Created with ink and camera effects). This carried over to other Tsuburaya shows such as Fireman and Kaiki Daisakusen as well.
 * And the fox.
 * Another Alan Ball show, Six Feet Under, also had an amazing title sequence. It was created by Digital Kitchen, the same studio that made True Blood's.
 * The 1985 version of the Twilight Zone opening.
 * Digital Kitchen, a design studio in Chicago, has created some fairly well known (and amazing) title sequences including:
 * Dexter depicts the morning routine in such a way that breaking eggs, making orange juice and putting on a t-shirt all look like acts of violence, all set to a jazzy tune. Awesome doesn't begin to cover it.
 * Six Feet Under
 * True Blood has a very creepy opening that includes things like kids eating strawberries cut so that it looks like they're feasting on raw meat. Alan Ball seems to love the studio.
 * The first season of Wonder Woman
 * Fringe uses creative changes in its title sequence to reflect the theme of particular episodes. When the episode is set back in the '80's and deals with the characters' backstories, the titles use an '80's font and digitised theme music, reminiscent of the period. When the episodes are set in the alternative Fringe universe instead of our own, the titles are set in a red background, as opposed to the normal blue. Perhaps unsurprisingly (you don't expect a TV channel to show that level of detail, do you?), SKY in the UK appears unaware of these conventions and frequently uses the wrong colour background card prior to ad breaks, which can leads to no small confusion.
 * American Horror Story has a gorgeous title sequence that managed to also have several Chekhov's Guns hidden in it.
 * The US remake of Shameless.
 * The X-Files

Western Animation

 * Batman the Animated Series featured a specially made fight scene between Batman and a couple of crooks. Unique in that it had no credits, no title, and no sound effects except for the coordinated Theme Music.
 * Superman the Animated Series was originally going to have a similar stylised opening showing Superman demonstrating all his powers. In the end time constraints meant that only a short sequence of him flying was included, and a more traditionally animated opening showing him growing up was made.
 * Justice League had a highly stylised opening that showed all the main characters using their main powers or abilities, with weird lighting effects.