It depends. If you're looking for on-going series, I'm not aware of many. I know Disney publishes stuff for kids, and I think the Archie series is still going.
The only other one I've seen is
Owly, by Andy Runton. This is an all-ages comic that appeals to kids with cute pictures and wordless expressions, but has enough plot and character to appeal to most adults. It's a story of an owl who is lonely because all the other animals and birds are afraid of him. One day he rescues a worm who's in trouble and takes care of him, and the worm decides Owly's not a bad fellow after all. After a trip home for the worm, the two become friends and go through a series of adventures together. If you're looking for explosions and block-buster-type action, this isn't it, but it's a nice introduction to story-telling through progressive pictures and facial expressions.
Another wordless story series is
Gon, by Masashi Tanaka. I think they are out of print at this point, but there were several:
Gon on Safari,
Gon Underground, etc. Gon is the world's toughest (and smallest) dinosaur, and he stomps his way from one confrontation (and friendship) to another.
I'm not sure if this is a series, but I like the trades I've seen of James Robinson's
Leave it to Chance. The stories take place in an alternative Earth where magic and science both work. Chance Falconer wants to become a paranormal investigator like her father, but that sort of thing supposedly only happens with the males in the family. Yeah, right. Chance is a smart girl who gets herself into and out of trouble, along with a cast of police officers, villains, and various supernatural creatures.
A one-shot book that's popular in my library with the under-10 crowd is
"It was a Dark and Silly Night", which contains a bunch of short comic-type stories from a variety of writers and artists (Neil Gaiman, Lemony Snicket, etc.). The stories are all unrelated, and the only thing they have in common is the beginning line of "It was a dark and silly night...."
I'm not sure if this is a one-shot or not, but I liked
Captain Raptor and the Moon Mystery, by Kevin O'Malley. This is an actual hard-cover book that wasn't originally a comic series. The main characters are all dinosaurs, but they have space ships and advanced technology, and they go to their planet's moon to investigator a strange light that flashes across the sky.
Depending on the reader, I can sometimes recommend the
Usagi Yojimbo series about the former-samuri-turned-ronin rabbit Usagi. The characters are all animals, and the violence is pretty bloodless. The stories and "chapters" are short and the art is good. Usagi generally wanders from place to place, fighting villains and helping non-warrior folk.
I'm seeing a bunch of "cine-manga" books published that are basically television shows and cartoons turned into comics. The titles include Kim Possible, Spy Kids, some of the Disney stuff like
Finding Nemo, etc. Barnes and Noble Online lists a bunch of them. I'm not sure how many are on-going. (There's even one that's biographies of NBA players!)
I happen to like the Asterix series, which I first read many (many) years ago in high school French class. Most of the violence is "cartoon violence" and more funny than alarming. There are a bunch of plays on words and some social/political humor that probably only adults would get, so the title can be read by pretty much anyone. Asterix, of course, is a French "barbarian" dealing with Roman invaders and other problems back in (or somewhere around around, anyway) 50 B.C.
I'm also going to send you to this thread at Millarworld on
Good Comics for Kids. Someone there asked a similar question, and received various answers. Let me know if the link doesn't work, and I can post a list of recommended titles.
I expect to be visiting my LCS next week, so I can ask them for recommendations as well.