Look at the major American animated films being released this year and you'll find almost all of them are 3D computer animation (Laika's Kubo and the Two Strings being a rare stop-motion exception). In contrast, 2D hand-drawn animation still makes up the majority of Japanese anime. Computers are still part of the production process. Almost every anime since 2000 has used computer coloring (Sazae-San was the last hold-out to switch from traditional cels to digital paint in 2013). The use of full 3D CGI, however, is much more limited due to a mix of budgetary issues and aesthetic tastes. Backgrounds and machines are frequently modeled and animated digitally, but anime with 3D CGI character animation are much rarer, and with a mixed success rate.
The '80s: The Birth of 3D Animation
...the technology wasn't quite there to pull off the effect they wanted. To be fair, this was VERY early in the development of computer animation. Tron, the first movie to utilize CGI for longer than a few seconds, had just come out the year before. But Tron's visuals have aged better than Golgo's helicopter because it worked with the limitations of early CGI, setting the action inside a video game and going for a relatively abstract style.
The stylistic influence of Tron can be seen in 1984's Lensman movie, which featured computer-animated spaceships.
The '90s: Major Advances
CGI remained a rare novelty in anime throughout the '80s and into the early '90s. In 1995, however, the same year that Toy Story, the first fully-CG animated film, was released, Japan had its own technical ground-breaker in Ghost in the Shell. Considered one of the most impressive mixes of traditional and computer animation at the time, the 3D effects haven't aged as well as the hand-drawn work (in 2008 Production IG released a reanimated version to raise the 3D animation quality to that of the 2004 sequel), but Ghost in the Shell made 3D CGI viable for anime production. Soon TV anime such as Escaflowne and Cowboy Bebop would include 3D CG background effects.
Blue Submarine No. 6
Gonzo made a name for itself spending big budgets on anime with CG backgrounds and mechanical effects, starting with the OAV series Blue Submarine No. 6 in 1998. They even called themselves "Gonzo Digimation" up until July 2004, after the premiere of Samurai 7, at the time the most expensive TV anime ever made due to its extensive 3D mecha animation. Budgets fell at Gonzo since, and they're no longer considered the cutting edge of animation technology.
Even the traditionalist Studio Ghibli experimented with 3D effects. 1997's Princess Mononoke used computer animation to deal with the complicated tentacles of the cursed boar gods. They'd continue to integrate 3D CG elements into their films, most notably the castle in 2004's Howl's Moving Castle before Miyazaki decided to shutter the studio's CG department during the production of Ponyo.
The 21st Century: Common Uses of 3D CGI
So how is 3D CGI best used in anime? As this making-of video for the 2004 film Steamboy shows, it create extremely convincing backgrounds and vehicles that blend in with hand-drawn characters while allowing greater freedom of camera movement.
As CGI becomes cheaper, effects animation can achieve more complexity, allowing for 3D mecha battles like those in 2010's Mobile Suit Gundam Unicorn. "Cell shading" rendering provides cartoony outlines and sharp colors that allow the 3D visuals to match with the 2D animation.
Fully Computer-Animated Anime
Final Fantasy: The Spirits Within
Fully computer-animated anime remain fairly rare. Japan's 3D animation talent is primarily focused on video games, so it makes sense many 3D computer-animated anime have been connected to video games (in the case of 1999's cheap and fascinatingly bizarre Gregory Horror Show, the anime came before a video-game, but it felt like watching a first-person video game cutscene minus the interactivity). Square attempted to get into the filmmaking business with the 2001 American co-production Final Fantasy: The Spirits Within, which attempted photorealistic rendering of human characters but fell short, falling into the Uncanny Valley for many viewers, and ended up a bomb.
Final Fantasy Advent Children
2005's Final Fantasy VII: Advent Children, in contrast, went for a similarly detailed more stylized look with designs closer to traditional anime, and was better received. Films such as 2009's Oblivion Island and 2014's Stand by Me Doraemon have also translated traditional anime designs into full 3D to acclaim.
Knights of Sidonia
Some 3D CG anime go further in attempting to replicate the look of traditional anime with full cell-shading. The various movies of Appleseed have been done in this style, while the 2014 Netflix series Knights of Sidonia is probably the current most prominent example of a fully cell-shaded anime. But does Sidonia and other cell-shaded anime sometimes go too far in attempting to imitate traditional animation? The most controversial issue is frame rate. Traditionally-animated anime tends towards a lower frame rate, but various drawing techniques and modulation of frame rate make the more limited animation flow naturally. Shows like Sidonia and this season's Ajin also use a lower frame-rate, but due to the nature of 3D modelling aren't able to utilize the same techniques that lessen the choppiness of traditional anime and thus appear unnaturally choppy. If 3D computer animation is to grow in Japan as it has in America, its best bet is to embrace the individual strengths and work around the weaknesses of its own medium rather than merely to slavishly imitate 2D animation and pale in comparison.