Someone else wrote off Frank Miller, but I have an enduring love for The Dark Knight Returns. Just pretend you’re still in the 80’s.

And just so I don’t come off as a complete psychopath for recommending a series that contains an entire issue of literal monkey sex, my favorite light-hearted comic series is Impulse. Specifically the early issues by Mark Waid. Aimed at kids, utterly charming. Alas, they only put out one collection, so it’s impossible to read without hunting down individual issues.

From Hell and The League of Extraordinary Gentlemen are both great (if you've ever seen the movies, please, please don't let that put you off).

A little bit further off the beaten path, at least in the US: The Ballad of Halo Jones is an early SF series by Moore that's just wonderful, and A Small Killing is a graphic novel from the early nineties that's subtle and uses the medium so cleverly.

As a primer for how comics work, what the medium can do, how a comic strip manipulates time, read Scott McCloud's Understanding Comics.

Grant Morrison's brilliant, very influenced by Borges, lots of metafictional stuff. Start with WE3, move onto Doom Patrol and The Invisibles. He's done a fantastic, self-contained, 12 issue run of Superman, All Star Superman.

Paul Pope's THB (difficult to track down, but he's working on a reissue) and 100%.

The Adventures of Luther Arkwright – an epic parallel universe sexy, action-packed series written and drawn by Bryan Talbot that was at least twenty years ahead of its time. His more recent Alice in Sunderland and Grandville are also fantastic.

Talbot and Kevin O'Neill, artist on League of Extraordinary Gentlemen, both did early work on Nemesis the Warlock, which is well worth a look.

Oh, and Death Note. An incredible, twisty-turny series.

(And as a general tip: if you're looking for stuff from the UK, try bookdepository.com, where everything's below cover price and there's free international shipping).

…

As multiple people have suggested Sandman, I'll go for a slightly more obscure Neil Gaiman comic, 1601. It's a re-imagining of the Marvel universe of superheroes (and villains), but set in 1602. It's great, not to be missed if you were ever into superheroes.

I love the graphic novel as a medium for storytelling, and that it's getting increasingly acceptable as an art form and not "just" pop-culture chic. This post made me so happy.

My recent favorites include:
Y: The Last Man

Fables (fairy tale characters in real life).

Midnight Nation by J. Michael Straczynski (of Babylon 5 fame).

Fray by Joss Whedon. Think Buffy in the far future.

Transmetropolitan by Warren Ellis. It's extremely graphic in every sense, but there's a strong thread of heart and storytelling–and absolutely stunning art–that makes it all worth it.

The Walking Dead, if you like post-apocalyptic/horror tales

Oh, and if you're into the more traditional superhero comics, run, don't walk, and pick up any of the recent Green Lantern titles.

A few that came highly recommended to me, but I haven't had a chance to read yet:

Pride of Baghdad by Brian K Vaughn (stray animals in the midst of the Iraq war – story is told from their perspective)

Lions, Tigers & Bears – an indie comic that I, sadly, know very little about, except that the comic store owner was very enthusiastic about it.

Astro City (gorgeous art deco style drawing, and supposedly a good story to boot)

Queen & Country – James Bond-ish spy thriller

Looking forward to everyone else's recommendations. I can never have too many comics!

I also highly recommend the first volume of Thor, by J. M. Straczynski. I know superheroes aren't considered to be very literary, but J.M.S. really pushes his readers to ask questions about what faith and the gods mean for humanity from the first issue.

I started them because I didn't see enough diversity in comics. I've got three webcomics up now (all paranormal) and hopefully one will be a hit with readers!