Our heroes are two socially challenged gamer siblings who suffer panic attacks if exposed to the outside world and/or separated from each other. However they're also the world's greatest gaming geniuses. One day, they play chess with a god who's so impressed with them that he spacewarps them into a world in which everything is decided by games. That does indeed mean everything, from war to the justice system.
Our heroes take on the entire world, game by game. It's fantastic.
The most basic reason for this show's greatness is its intelligence. The games are really tough and every opponent pushes our heroes to the wire, but they still keep winning and the show gets deep into all their clever tactics and strategies. They're geniuses. What's more, they're not geniuses in name only (i.e. as is often the case when the writer of a book or TV show needs a character to be a genius but has no idea how to demonstrate this). Here, the writing's backing it up. They are terrifyingly but convincingly smart. They outthink not only around their opponents, but also probably the audience watching the anime.
The show plays fair. It tells you the rules. You're being invited to spot the loopholes and angles, just like Sora and Shiro. It's possible that you might, but it's quite likely that you'll get blindsided by something you hadn't spotted at the time but is logical in hindsight.
We thus have terrifying games, full of brilliantly clever tactics and played for the highest possible stakes. Win a game in this world and your pre-agreed request will come true, even if this means wiping your opponent's memory or rewiring their brain. That's scary. The games themselves are generally familiar, by the way, but with a twist. The chess game we see, for instance, has self-aware pieces that may or may not choose to obey their commander's orders.
That's all good, but in itself could theoretically be dry. The reason it's not is the characters.
Our heroes, Shiro and Sora, are awesome. This is partly because they're pathetic. Take them outside into the streets of Tokyo, for instance, and they'll turn into terrified, drooling, colourless blobs that will crawl away and hide. Sora is a perverted virgin (although he can't be as bad as he seems, since he must have had a million opportunities to abuse the games and order female opponents to have sex with him).
They're also arguably closer to being villains than heroes. Their machiavellian self-obsession and genius for manipulating others would have made them fantastic antagonists, so for instance I think they're pretty much unlikeable in ep.1. Sora is childish, arrogant and gloating, while Shiro interacts with the world as if it's a mathematical equation.
At the same time, though, give them an enemy and they're magnificent. There's no foe they won't take on. Humanity is ranked sixteenth out of the sixteen intelligent races in Disboard, with the others all having magical powers, superhuman speed and reflexes, etc. Mankind has none of that. We're weak... which is then turned into a virtue by Sora's brilliant speech in ep.4 about weakness being our strength. (What's more, he's right.) Mankind has been crushed. We've been all but pushed off the map, with the king having bled territory as he lost game after game against superhuman opposition. For Sora and Shiro, making themselves king of all mankind is just the first step in their plan to take over the world.
Their ambitions even extend beyond that, in fact, but what's cool is that you'd bet money on them achieving it all. They don't have a nanosecond's doubt, obviously. They'd bet their lives for a bluff, which is part of what makes them such fearsome opponents.
The supporting cast are fun too, by the way. Stephanie is the old king's serious-minded daughter and basically the straight man (and butt monkey) to Sora and Shiro's irresponsible-looking antics. Kurami is a ruthless, cynical enemy, but also devoted to Feel Nilvalen in what's either an intense (but platonic?) lesbian relationship or a mother-daughter surrogate.
The art is wild and almost hallucinogenic, with red-lined art and oversaturated colours. I liked that too. The only thing I didn't like was the occasional fanservice, not because Shiro is eleven (although she is) but simply because the show's otherwise so strong and focused that girls' bath scenes didn't fit. (It's possible to find online dislike of Shiro's panty shots, for instance, but the art's too stylised for these to have any chance of seeming sexy, or indeed for Shiro's age to be anything but an informed attribute.) That said, though, I laughed at one episode's parody of anime Barbie doll nudity, when Sora deletes the girls' clothing during a magical game only after first erasing their naughty bits.
Incidentally, there's another anime and light novel series about children being transported to a world that decides everything with games, called Problem Children Are Coming from Another World, Aren't They? I might watch that too, although I expect to prefer No Game No Life.
Summary: it's great. Sora and Shiro are deliciously arrogant, amusingly pathetic and the best damn magnificent bastards you'll see in a long time. The world of Disboard is big and bad, with entire races (the Flugel) created as weapons of mass destruction and the one we meet having once practically committed genocide. (She's sexy, by the way.) When can I expect a second season?