Yato is a god. Only problem is, he's such a low-level god that he can't even afford a shrine! He collects and saves 5-yen coins by doing odd jobs and granting people's wishes in the hope of becoming the most well known deity in Japan.
Enter Hiyori, a high school girl with a thing for mixed martial artists. She nearly dies saving Yato's life, with the result that she can switch between the physical and spiritual worlds. She gains some supernatural abilities, including being able to see the "phantoms" that haunt the depressed and anxious, but they can see her too. She hires Yato to solve this problem and the two become friends.
Rounding out the trio is Yukine, the spirit of a dead teenage boy who becomes Yato's weapon--literally. Gods turn pure spirits into swords and such in order to fight phantoms; in return, the spirit regains its physical form (at least to an extent) and some of its ability to affect the world. But Yukine is bitter and increasingly resentful about being used, not to mention being dead, and that's when the phantoms begin to circle. Yato and Hiyori need to figure out how to help their friend while granting wishes, dodging angry ghosts, and avoiding other gods that have it out for Yato.
This series sucked me in immediately with its characters. Yato is adorably determined and hilariously ungodlike (at least most of the time). Hiyori is one of the most refreshingly well-rounded anime ladies I've seen in recent memory. She's kind and intuitive but she's no doormat; she has her own hobbies and fears along with a surprising amount of strength. Yukine's slow descent into darkness is as frustrating as it is understandable, and you feel for him even when he's at his most difficult. The side characters are a bunch of fun as well, from an infinitely perky goddess of chaos to Yato's creepy as hell ex-weapon. Even the dark episodes have their humorous moments, and with only twelve episodes (all on Netflix, btw) it's an easy series to get through in one sitting! Noragami brought me a lot of joy and more emotional moments than I was expecting; I'd definitely recommend it.
Psycho-Pass takes place in a future where everyone's mental state can be chalked up to a number, which is then used to judge their criminal potential--whether they are worthy of being hospitalized, jailed or even killed. The weapons used by police inspectors scan that number, or "crime coefficient", and, depending on how high it is, set themselves to "stun" or "obliterate". The enigmatic Sybil System performs these scans and thus keeps the peace of this society intact.
Tsunemori Akane is an idealistic but determined young inspector. She is partnered with the very by-the-book Ginoza, who has a massive chip on his shoulder and a relentless worry for his own crime coefficient. Their subordinates are Enforcers, unusual individuals who have been judged "latent criminals" and do the inspectors' dirty work for them. One of them, Kogami, is an ex-inspector himself with old-fashioned ideas about detective work. He's obsessed with capturing a charismatic, manipulative murderer who escaped his reach years ago, and when a viciously familiar serial killer appears on the scene, Kogami sees his chance.
The first season is a brilliant feat of dystopian world-building, surprisingly thorough characterization of even side characters, and a cat-and-mouse game between the officers and the criminals filled with twists and turns. The relationships and moral conflicts between characters are complex and carefully built up over the course of 24 episodes. Psycho-Pass also doesn't shy away from the darker aspects of its imagined future. Citizens are perpetually concerned with keeping their crime coefficient down, which they manage through excessive therapy and medication, as even someone with severe depression can be marked a latent criminal. The ever-optimistic Akane butts heads with cynical Kogami and others around her in her determination to protect the law, but she is ultimately forced to confront the nature of the society she is defending--and the nature of the Sybil System itself.
I can't sum up the second season without spoiling the first (the latter is on Netflix while the former is only on Hulu at this point), but it is just as thought-provoking and amazing in its many shades of grey. This series made me think, made me wonder, made my cry and sit at the edge of my seat. Psycho-Pass isn't for the faint of heart (or stomach), but it's really a spectacular ride.
So, there we are--my overlong opinions on anime that no one asked for. Seriously though, if you're looking to dip into some really good series, these two are a great place to start.