Here they are:

1. Dark manga, like Oyasumi Punpun, Homunculus and Akira. I'm fine with either character-driven stories like Oyasumi and Homunculus or plot-driven stories like Akira, as long as they involve moral ambiguity/the nasty side of humanity.

2. Serious romance stories or stories involving romance, in which there's unrequited love or the couple faces lots of obstacles. Examples - Hachimitsu to Clover, Mars, Bokura ga Ita, Kabocha to Mayonnaise. I would be especially interested in stories where the person finds love but chooses not to have a relationship due to a different passion or whatever other reason.

3. Weird manga that, I guess, defies genre a bit? And maybe has an element of surrealism? Things like Nijigahara Holograph, Tamashii No Futago, Bernie's Drawing Diary and Palepoli (but Palepoli is too extreme, I want something with a plot).

Qualifications:

Must be complete.
Seinen, josei, shoujo and shounen are all okay.
Unless it will totally blow my mind, I don't want a fantasy story. Supernatural or sci-fi are okay, I just want it to take place in this world.

Thanks people

you might want to make your lists public so that people do not recommend things that you have already read.

1
the first one that i thought of was Ajin which fits very well however it is far from complete.
some that are complete that fit are
Death Note (though you might have already read this one)
Hanged Doll and Superior Day -these two are webtoons (not sure if you mind)
Phantom - Requiem for the Phantom -not sure how the manga of this one is i watched the anime

not sure about number 2, i haven't read any of the examples but maybe Hana to Akuma but this one has some fantasy in it, though its mostly set in our world in the past.

for 3 i would definitely say Annarasumanara it has a surreal feel to it and its one of the best ones i have ever read (its a webtoon again though)

Last edited by Gale1 at 5:30 pm, Jul 7 2014

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Like educational manga? take a look at Ed U. Manga's list of manga here

1) Have you read BAA? Maybe Deadman Wonderland. Obviously FMA too.

2) Natsu no Zenjitsu, Yesterday o Utatte (warning: slow), Emma, Solanin

1. Helter Skelter
Umibe no Onnanoko

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Favorite Art:
Oboreru Knife
19 Days
Gangsta

3. Annarasumanara

Thanks for the suggestions. I have now made a list. I was too lazy before, I stored the data previously on my phone, but I can see how a list would provide less rage in the people offering suggestions

Gale - I have read Death Note and Hana to Akuma and really liked them both. I'll check out the webtoons - have never tried a webtoon before, so I dunno whether I like them. Phantom looks good but it's not scanlated, so I'll have to add to my 'maybe buy someday' list.

Sunder - I've read Deadman Wonderland. BAA looks interesting, I'm going to try it out. I still haven't read FMA since I loved the first anime and don't want to see a different ending (sacrilege, I know!!) but I will read it someday! Read Solanin, unsure whether I would be interested in Emma, but I'll give it a try. Others aren't fully scanlated but I'll record the names to try when they are.

Sphear - Read Helter Skelter. I haven't read Umibe no Onnanoko yet but it's definitely on my to-read list.

1 :
Bokurano Survival manga with giant mechas. Very dark.
Duds Hunt The premise isn't original : a survival game, but it's short, the MC is an evil bastard and the ending was good (although kinda predictable).
The Embalmer
Parasyte You've probably already read it.. if not, do so.

[ 1 ] Doctor Du Ming
[ 1 ] Gankutsuou- sci-fi, set in the future
[ 1 ] [ 2 ] Transient Flower
[ 2 ] Miman Renai
[ 3 ] Coelacanth

Second these two for the same reason as above.

3
Maybe Kono Sekai no Owari e no Tabi and Marie no Kanaderu Ongaku?

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1
Aku no Hana
Onani Master Kurosawa


2
Usotsuki Mi-kun to Kowareta Ma-chan: Totteoki no Uso?

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Ran to Haiiro no Sekai is a really good fantasy manga.

1:
Not Simple
Motto, Ikitai...
K no Souretsu (arguably worth listing under your third type)

3:
Plastic Girl (FURUYA Usamaru)
G (KOIKE Keiichi)
Travel (and anything else by Yokoyama Yuuichi)
Various oneshots by Tsuge Yoshiharu (The Deadly Dried Squid Technique, Screw-Style...).

And I second Journey to the End of the World.

So many great suggestions! Thanks everyone, I'm going to try 95% of them.