Oo everyone in this thread has good taste
My favourite that I like to peddle to people is ACCA: 13-Ku Kansatsuka (13 Territories Inspector). It's a mystery, but not just any mystery-- a political one! There's a rumour that someone is trying to overthrow the government, and no one knows how much of it is true, or who started it. It was really fun seeing a mystery genre that wasn't the standard murder mystery or like stolen valuable or whatever. Also the writing was fabulous. The mystery has a lot of layers so you get the fun of figuring some things out, and also being totally blown away by other parts.
ALSO ALSO the music is great. I almost never notice soundtracks, but I did for this show. So good.