My Favorite Non-Superhero Comicbooks! 

Saga

Romeo and Juliet in space! Except Juliet doesn't kill herself, but instead kicks ass and Romeo is a pacifist! Alana and Marko is from two different races which is fighting a seemingly never-ending war! They fall in love and make a baby! Now the entire universe is after them, enemies and family alike!

A really grown up comic (typically Image Comic) with moments you really shouldn't show the common stranger beside you on the bus.

Low

Mankind has fled to the deepest oceans to escape the merciless radiations of the sun! They sent robots to space to find a different world to live in. We follow the story of a mother in desperate search for her daughters who were taken from her, doing anything she can to find them.
A beautiful comic by Rick Remender who also made Tokyo Ghost and Black Science!

Rat Queens

I'm just going to copy Goodreads description of this comic, because they covered it pretty nicely:

Who are the Rat Queens?

A pack of booze-guzzling, death-dealing battle maidens-for-hire, and they're in the business of killing all god's creatures for profit.

It's also a darkly comedic sass-and-sorcery series starring Hannah the Rockabilly Elven Mage, Violet the Hipster Dwarven Fighter, Dee the Atheist Human Cleric and Betty the Hippy Smidgen Thief. This modern spin on an old school genre is a violent monster-killing epic that is like Buffy meets Tank Girl in a Lord of the Rings world on crack!

(https://www.goodreads.com/book/show/20299683-rat-queens-vol-1?ac=1&from_search=true)

Giant Days

3 girls in college, doing college stuff and having collage problems!
Sometimes it's just nice reading a feel good comic, without the supernatural and fantasy theme. I love the silly illustrations and slow-moving story!

Clive Barker: Next Testament

God has been trapped and hidden away. Now he's back, and he reeeeally doesnt like what he sees.
This is a really trippy, wacky type of comic, but something just makes you like it. May be the colors, may be because it's a Clive Barker story, or maybe it's the typical story of humans being persistent and striving to stay alive.

Cat Tax!

Edit: Thank you guys for all the nice comments, recommendations and the huge love toward Saga!! You guys are great! And The Frontpage!!!!!! Thank you!
I also want to point out that this list holds a lot of new comics, but I do like a lot of the older comics too, like Fable, The Cape, Blacksad, Walking Dead, Sandman and such, but these are the ones I really really like ;) I got introdused to comic books really late, because I'm from Norway and we only read Donald Duck and Conan here as kids, but I'm working on getting around in the comic world, so again, thank you for your recommendations!