These are gen recs especially starring the Weasleys. If any fics about any other Weasleys turn up, they'll be included here also. Consider them pure gen unless otherwise noted.
Arthur and Molly
Bill and/or Charlie
Percy
Fred and George
Ron
Ginny
Fics starring multiple Weasleys
And Let You Go (PG) by gmth. The Burrow is empty, even when it's full. Very sad. One shot.
Clockwork (PG) by xylodemon. Molly waits up for Arthur to come home. Lovely portrait of her. One shot.
Father and Son (G?) by IsabelA113. Mention of canon Ginny/Dean. After OotP, Arthur and Ron have a talk. One shot.
Lux Aeterna (PG) by narie_the_waitress. The summer before OotP, Molly Weasley takes a nighttime stroll through the halls of Grimmauld Place. She's well characterized in this fic. One shot.
Missa (G) by Mirabella. A series of post-War vignettes. Bittersweet. One shot.
Spark (R?) by Jane St Clair. During the First Voldemort War, Molly Weasley does what she has to do. I love the ways in which family is portrayed here. Angsty. One shot.
The Wizard From Outer Space (Meta Remix) (PG) by AJ Hall. This is a remix of a drabble written by Halrloprillalar which is quoted in its entirety at the beginning of the fic. Arthur Weasley stumbles into a con. Written from the POV of one of the fen, it's quite reflective. One shot.
Wordless (PG?) by penknife. Arthur Weasley thinks about his children, Percy especially. Awesome characterization. One shot.
Adventures in Curse-Breaking (MOM1, by Godric's Hat ratings) by Lady Narcissa. Bill Weasley goes on an assignment. Clever. One shot.
Charlie Weasley and the Vampire-Vanquishing Solution (G) by After the Rain. Snape recieves an unexpected visitor to one of his potions classes. Funny, and thoughtful toward the end. One shot.
Not Without My Reasons (PG) by ladyjackyl. Bill and Charlie go to visit their recently fired brother Percy to try to convince him to come home. As one of it's reviewers states, it gives justifications for Percy's actions without making excuses. One shot.
Once Upon A Time, In Egypt (PG-13?) by Josan. Offhand mention of m/m; otherwise pure gen. Bill Weasley tries to convince former Hogwarts professor Severus Snape to come to Egypt and work on a burial dig. It's excellent all around. Long one shot.
Perfection (PG) by Cairnsy. Bill and Percy have a bit of a talk. One shot.
A Little Bit of Music (PG-13) by rhoddlet. When Percy was younger, he was bit by the photography bug. This fic is creepy and surreal. One shot.
Caves (PG) by rhoddlet. A creepy, excellent Evil!Percy story. One shot.
Dust (PG-13) by Amber. I'm usually not a big fan of 9/11 tributes, but this one is unbelievable. Hogwarts is a disaster zone, and Percy just keeps on working away at it. Sad. One shot.
Nice Work If You Can Get It (PG) by Kay Taylor. Hermione gets an office job for her Muggle Studies class. The end turns it into a completely different fic. One shot.
Not Without My Reasons (PG) by ladyjackyl. Bill and Charlie go to visit their recently fired brother Percy to try to convince him to come home. As one of it's reviewers states, it gives justifications for Percy's actions without making excuses. One shot.
Dream Deferred (PG-13) by Risti. Includes a Percy/Penelope breakup. Percy reviews his prospects after the events of OotP and decides where to go next. WIP, to be novel length.
Perfection (PG) by Cairnsy. Bill and Percy have a bit of a talk. One shot.
Sir Percy and the Dragon (PG?) by Kest. There's an H/D-ish subplot. Percy is sent by the Ministry to kill the dragon Marcus Flint, with aid from several animals and inanimate objects who we know. It's weird, but oddly appealing. One shot.
Unlocking Door (PG-13) by The Treacle Tart. After OotP, Percy Weasley is reassigned to the Ministry of Magic's Safety Deposit Facility, where many of the Pensieves stored during the First Voldemort War were forgotten. One day, he finds the key to a vault that he somehow never noticed before. An interesting idea. One shot.
Wordless (PG?) by penknife. Arthur Weasley thinks about his children, Percy especially. Awesome characterization. One shot.
The Wrath of Loki (MOM2) by mosylu. Percy lies in the hospital, near death, and where are the twins? One shot.
Frayed (PG) by Sarah H. George Weasley realizes that he's growing apart from his twin. Creepy. One shot.
George Weasley, Shadow Man (PG) by Morrighan. Fred has given George a diary for his birthday, and he writes in it. Thoughtful, a bit angsty. One shot.
Missa (G) by Mirabella. A series of post-War vignettes. Bittersweet. One shot.
The Secret of Our Success (PG-13) by YayCoffee. How Fred and George came into possession of a certain Map and learned how to use it. Cute and funny. One shot.
The Taste of Smoke (PG-13) by Tani. There are some things that cannot be forgotten. For George Weasley, the taste of smoke is one of them. One shot.
The Wrath of Loki (MOM2) by mosylu. Percy lies in the hospital, near death, and where are the twins? One shot.
A Run in Tights (PG-13) by cawti. Pure gen, except for leering. The Trio has to get to Platform 9 and 3/4 in time to catch the Hogwarts Express, but Death Eaters are everywhere. So how do they elude them? By dressing as Muggle schoolgirls, of course. This fic is cute and fun, but still manages to show the danger they're in. Long one shot.
Bottom of the Well (PG) by rhoddlet. There's Parseltongue being spoken at the Burrow. Creepy. One shot.
Father and Son (G?) by IsabelA113. Mention of canon Ginny/Dean. After OotP, Arthur and Ron have a talk. One shot.
fictionkill (G) by fivil. Harry still remembers some things. Sad. One shot.
Golf Balls (G) by Wraith 11. Harry, Ron, and Ginny come up with a new game. Funny, if a bit mean-spirited. One-shot.
Gryffindor Colors (PG) by Dreamcoat_mom. Minerva has some trouble getting to sleep after Ron's big Quidditch match. Cute. One shot.
Harry Potter and the Muggle Toy Store (PG) by MadisonS. Includes a reference to H/Hr. Harry, Hermione, and Ron go to a toy store and see themselves. I like Elsie. One shot.
Heart Stopped Breathing (PG-13?) by durendal. The main text is a simple account of Ron buying the Sneakoscope in Egypt, but a very different story is going on between the parentheses. Disturbing, in a good way. One shot. The Remix, written by Iosselen and rated PG, turns this story inside out, and the last line is a kicker.
How the Story Ends (PG-13) by Sioniann. Includes some references to het and slash. Neville can't remember, so he has to make up memories. Beautiful and lyrical. One shot.
I Have a Rendezvous with Death by Paracelsus. Includes past R/Hr. Harry puts together a devious plan to defeat Lord Voldemort. Intricate plotting. This is the prequel to And Miles to Go Before I Sleep, but can be read independently. Novel length fic.
In Your Own Nature (G?) by VoxMaille. Plays with the idea of Ron as a Seer, and how he considers it. One shot.
Jehu (R) by Nathan Stack. Includes one or two references to relationships. Harry is out hunting Draco Malfoy with his crew. A wonderful, fantastical takeoff of Moby Dick, though you don't need to have read that book to appreciate this fic. One shot.
The Lost Boys (PG-13) by Ishafel. A look at the scars the wizarding world leaves on its children, specifically Draco, Ron, and Harry. Angsty. Chaptered fic.
Midnight By the Weasley Watch (PG) by excessivelyperky. Includes flirting. Ron wins the Hogwarts Chess Tournament and gets to play in an international tourney, accompanied by Snape. The descriptions of the chess games are fabulous. Novel length.
Midnight Confessions (PG) by Diricawl. It could be pre-R/Hr, or it could just be two friends talking and bickering. Ron and Hermione talking in the Hospital Wing after the MoM battle. One shot.
Missa (G) by Mirabella. A series of post-War vignettes. Bittersweet. One shot.
Mourning Rites (PG) by Hijja. The Dursleys take care of Harry's old room after he dies. Sad. One shot.
The Pop Goes the Weasel Trilogy (PG-13?) by durendal. Very humorous. Ron is a dopehead at 26. Lucius gives him a job in his organization. Draco is his secretary. This fic has a lot of great lines in it. Three fics; the entire thing is novel length.
Things That Don't Exist (PG-13) by shadoed. A Muggle notices a striking girl in the library. Short one shot.
The 'A Day In the Life Of...' Series (G and PG) by Angela Boyko. Small bits of random shipping, mostly in the Ginny chapter, but mostly gen. An average day for several HP characters, including all three Malfoys, Rita Skeeter, Dobby, and Ginny Weasley. Amusing, especially Dobby's entry. All one shots.
After (PG-13) by Serpenthe. Harry loses, and Ginny must decide what to do next. Excellent darkfic. One shot.
Better Days (G) by Antosha. Femmeslashy undertones, but can easily be read as gen. Voldemort has been defeated, but a lot of people have died as well. Luna helps Ginny come to terms. Sweet. One shot.
Bottom of the Well (PG) by rhoddlet. There's Parseltongue being spoken at the Burrow. Creepy. One shot.
Disenchanted (PG?) by Arabella. After Tom's diary is disposed of, Dumbledore goes to have a talk with Ginny. One shot.
Empire of the Senseless (PG-13) by A. L. Milton. There's a past Ginny love affair, but otherwise it's pure gen. Grown up Ginny goes on some adventures. Very Pratchett-esque. Chaptered, likely abandoned WIP.
Golf Balls (G) by Wraith 11. Harry, Ron, and Ginny come up with a new game. Funny, if a bit mean-spirited. One-shot.
I Have a Rendezvous with Death by Paracelsus. Includes past R/Hr. Harry puts together a devious plan to defeat Lord Voldemort. Intricate plotting. This is the prequel to And Miles to Go Before I Sleep, but can be read independently. Novel length fic.
Just Words (PG), by Ayla Pascal. Ginny Weasley works for the Dark Lord, hunting down Mudbloods. Dark. One shot.
Pedigree (PG?) by crimsonclad. Pure gen, other than knickers stealing. Ginny is captured by Voldemort, but not for torture -- as a pet! Cute. One shot.
Secrets (PG) by Elizabeth Culmer. Canon Ginny crushing on Harry, and Tom feeds her some story about a lost love, but it's mostly gen. CoS from Ginny's point of view, including the reasons why she felt so isolated from the other students and some interesting thoughts about House politics and prejudices. Chaptered WIP, updated infrequently.
'Till I Wake Your Ghost (PG-13) by lifeinwords. Ginny has ink in her veins. Creepy, with great imagery. One shot.
The Very Secret Diary (PG-13 or R) by Arabella. Includes Ginny's crush on Harry, Ginny crushing on Tom and (I think) the canon Percy/Penelope relationship, but it's overwhelmingly gen. A reconstruction of Ginny writing in Tom's diary during CoS. It's just... definitive, really. Novel length.
And Let You Go (PG) by gmth. The Burrow is empty, even when it's full. Very sad. One shot.
Bottom of the Well (PG) by rhoddlet. There's Parseltongue being spoken at the Burrow. Creepy. One shot.
Chance Encounters (G?) by Arabella. The Dursleys go to London and drag Harry along, where he runs into a few people who will be important to him later in life. Cute. One shot.
The Familiar and the Family (PG) by Donna Immaculata. One tiny bit of Peter/Sirius. Peter's first few weeks with the Weasleys. One shot.
Fed, Dorge, and the Unwitting Babysitter (PG-13) by Trillian Black. The Weasleys hire a babysitter to take care of their youngest children. Hilarious. One shot.
I Have a Rendezvous with Death by Paracelsus. Includes past R/Hr. Harry puts together a devious plan to defeat Lord Voldemort. Intricate plotting. This is the prequel to And Miles to Go Before I Sleep, but can be read independently. Novel length fic.
Loosey Lippity Lessons (G?) by FernWithy. The five youngest Weasleys have lessons, with great doses of alliteration as Ginny learns her alphabet. Unrepentant fluff. One shot.