Trailer: Fan-made AMV
What it’s about: Karuta is a game of poetry and cards, native to Japan but little known elsewhere. Of course, this means that whoever reaches the top national rank is not only the best at the game in their country, but the best in the world. The tomboyish Chihaya learns of karuta from her friend Arata, a loner obsessed with living up to his grandfather’s legacy as a karuta master. She sees in the game a way to step out of the shadow of her sister, to find a dream of her own.
Why watch it: Like all the best sports anime, the actual game being played matters very little, and the nature of karuta makes audience predictions and strategising entirely moot. Instead, the show uses the game as a lens through which to examine the characters and their relationships to one another and to the game itself. All three protagonists are incredibly well-realised; I can only think of a handful of other anime with characters that seem so human: flawed, vulnerable, but capable of greatness. The art and character designs are utterly gorgeous and the soundtrack, while unobtrusive, ties the show together. This is a really, really strong show and all too easy to binge through.
Caveats: You don’t need to know anything about karuta going into it. It won’t affect your enjoyment at all. This is a very character-based show, so what happens in the plot is never as important as the way the protagonists react to the changing circumstances. There is no dubbed version available, and the nature of the sport makes it uniquely unlikely to ever receive one.
Themes: Finding a dream for oneself, and dedicating one’s life to its pursuit. Bonding through shared experiences.
Similar works: Ping Pong: The Animation
Trailer: PV trailer.
What it's about: A near-future cyberpunk procedural drama. A computer system has been developed that can automatically and instantly determine a person's psychological state and their likeliness to commit crimes. Its diagnoses are so accurate that society has used it to entirely replace the justice system. If your "psycho-pass" reading is too high, you're a threat to society and should be locked up. The story follows a police task force whose job it is to investigate the handful of cases that fall through the cracks of the surveillance state.
Why watch it?: It's a "dark" action-based story, and as an anime it's probably the most easily-translatable to a Western live-action drama that I've ever seen. It's an exploration of a deeply flawed utopia that expertly balances philosophical discussion with heart-pounding action. At the end of the show you're left to decide for yourself which, if any, of the various powers were correct in their actions and motivations.
Caveats: The first season is actually a pretty solid all-round show. That said, the literary references are a little pretentious, and the female protagonist is annoyingly naive in the first episode or two. The second season has been a bit more controversial, receiving criticism over its writing and overall direction.
Theme(s): "He who fights monsters should take care not to become one himself". The price of peace - how much individual liberty is it worth to maintain a stable society?
Similar works: Ghost in the Shell. Also, any of the other works by Urobuchi are a good starting point - Phantom: Requiem for the Phantom is the closest from a storytelling perspective, but Madoka Magica is much better and has received an almost ridiculous level of critical acclaim. Shinsekai Yori addresses many of the same themes, but it takes a very different approach. Outside of anime, Minority Report, V for Vendetta or Blade Runner.
Trailer: Dubbed version
What it's about: Hana, a 19-year-old college student, falls in love with a mysterious classmate, only to discover that he's half-wolf. The romance comes to an abrupt end when the man is killed, leaving Hana to raise their two children alone. The film follows the family as Hana struggles to bring up her kids without revealing their true nature, with few resources and no friends to lean upon.
Why watch it: It's a fantastic story, told well and with utterly believable characters. It's impossible not to empathise with each of the three protagonists who, despite their differences in personalities, worries, and desires, nonetheless share a familial bond that ties them all together. It's a story about family, and one that I think has an almost universal appeal even to those who might not otherwise watch anime.
Caveats: Very few. Even if you're not a fan of family dramas I would recommend giving this a try before discarding it.
Themes: Family.
Similar works: Try some of the director's other works, like Summer Wars or The Girl Who Leapt Through Time. For something with a more thematic similarity, try Usagi Drop.