What is so special about anime? The best is renowned for the quality of the animation, and sometimes full screen animation is used, unlike the two or three cel layer partial animation in the West. Some use partial CGI for effects such as clouds or water, or CGI produced from hand drawn originals - more attractive than pure CGI as it stands in 2011.
Anime probably started with the MGM Japan release of "Magic Boy" in 1959 - about the time Disney had their last movie cartoon hit for some time with 101 Dalmations. Due to the number of visitors seeking info on this film and the scarcity of web information, I have added my own Magic Boy tribute page (caution- 500k of images).
The Anime Cafe is a wonderfully crafted site well worth reading.
You will find that anime covers everything from animations for the youngest children to very adult animation- there are many videotapes / DVDs sold in Japan of animation that UK customs would seize and prosecute you for- so be careful if you import to make sure what you are getting! Note that you do not have the protection of BBFC certification on imported titles and some Japanese titles may - however innocent in cultural context - cause legal problems in the UK.
UK residents will have met with anime in such lovely programs as Moomin and Alfred J Kwak - both of which have excellent animation and stories.
My special favorites in the movies are "My Neighbour Totoro", now available both subtitled and dubbed into English on DVD;
Kiki's Witch Delivery Service and tv series such as
Maison Ikkoku (tenants of an apartment);
Kimagure Orange Road (love triangle, very sweet) and Uresai Yatsura -(bikini clad alien falls in love with womenising loser).
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Can't not mention Sailor Moon which is better in the subtitled Japanese than in the US TV
dubbed version, but even the dub is OK.
Of course as a human owned by a cat,
What's Michael is especially endearing- if only I could find videotapes with subtitles!
The Shaw family are greatly entertained by the comedy present in Marmalade Boy, and greatly
enjoyed both the tv series of Little Women [site down 3/13] and the movie from Toei.
Another beautifully animated show in the World Masterpiece Theatre series of anime, which included Moomins, Little Women, Nadia and so on - Romeo's Blue Skies, a Swiss story which seems not to have been translated to English. The story is over a century old, the anime is very attractive.
More recent discoveries I can recommend:
Mahoraba - Heartful Days;
Victorian Romance Emma;
REC;
Rose of Versailles.
Kanon is beautiful (genres moe, bishojo - it is an adaption of a PC game). Go for the 2006 version.
Eve No Jikan (Time of Eve) is a short series or film with an interesting SF/Android theme. There are interesting subtle anti-discriminatory comments.
Chobits is a trifle more mature than some others listed here, the BBFC gave it a 15 cert (15 for moderate sex references, 12 for fantasy violence), but it is great story telling. This is one for serious fans of SF, robotics, Blade Runner and so on. There is some humour. I liked it. My link is to the reviews on IMDB, which seem more about the show I watched than the negative reviews on specialist anime sites who really didn't understand the show.
I could not end without mentioning the 39 episode NHK TV series
NADIA (The Secret of Blue Water)- a long story arc, cute leading teenagers, conspiracy;
aliens; cute lion cub... superb story and good animation. Episode 5 requires a handy box of tissues. Wow. Gainax advise that the central
13 episodes, based on the island, were farmed out to Korea, hence differing animation and characterisation - they suggest that their 26 episodes can be watched on their own. (first 21, last 5 episodes).
Several strong similarities to a later Disney feature animation.
21k picture of Nadia
There are many other excellent titles. Discover and enjoy.