Here are a couple of games that are not strictly RPGs, but they are very cool and cooperative.
First up,
SHADOWS OVER CAMELOT. This board game is a jewel - every player is a knight of the round table, and all are working together to defeat the forces of darkness, so it's players VS. the game itself. Every single turn brings new bad events, and the knights must go on quests, together or alone, to turn the tide of evil. Quests are carried out on different areas of several different boards, by traveling and then playing various cards from each knight's hand. Even with everyone's combined efforts, the game is tricky to beat, BUT THEN, there is a fun (though optional) twist. ONE of the loyal knights just might be a traitor, working underhandedly to see that the forces of darkness win. So, a great, relatively easy to learn co-op game, and since the traitor is assigned in a random blind draw, it shouldn't matter to your kids. (They may indeed relish the chance to be the secret lone wolf.) I recommend for 3 or 4 players. Visually, I think it's the prettiest game I have.
A very similar style of play is used in Reiner Knizia's popular
LORD OF THE RINGS board game. This is also a co-op, players are one of the hobbits from the Fellowship. There are action paths through the entire narrative of the book (along paths of friendship, travel, fighting and hiding) and a simultaneous path along a corruption track. Carrying the ring draws the hobbits inexorably over the corruption line and closer to the Eye of Sauron. Slightly more complex than Camelot - it may be an advantage that in this game there are never traitors. Everyone is in it together here. Four players maximum, and it can take a couple of hours. There are several expansions - I recommend the "Friends and Foes" expansion, because it completes the location boards so that the game goes through the entire narrative line of the novel, and it adds some considerable additional challenge to the game, along with some fun new ways of playing. The whole thing is really pretty, too.
The big enchilada in this category is
ARKHAM HORROR, based on the fiction of H.P. Lovecraft. Everyone is an "investigator" in 1930s New England, trying to track down clues, weapons, companions and arcane objects to prevent one of the elder gods from awakening. Portals to other dimensions keep opening up all over town, and monsters are shambling (or flying) through the streets. I like this one, because the players are librarians, scientists, students, con men, etc. Just regular folk in cosmically bad circumstances. This one is VERY complex, with a bit of a learning curve - but it is cooperative, and the game itself is just downright evil in how badly it wants you to lose (either the game or your sanity.) There are now about half a dozen expansions available. Even the base game is HUGE and will take up your entire dining room table.
I play all of these, and I sometimes play 3rd edition D&D as well. I like the fact that these board games have a self-limiting time factor, even if they take a few hours. One thing I have always disliked about D&D is the ongoing nature of it. I like closure, and I've never been able to commit to showing up somewhere on any regular basis to participate in an endless campaign.
A couple more quickies - If your kids enjoy D&D and you want to try a board game along those lines, there is DESCENT, which is another gargantuan Fantasy Flight superbox game. It comes with tons of plastic figures and a modular board that can be re-arranged in endless ways. Like D&D, this is guided by a dungeon master, but in this case, the DM is actively, and aggressively, trying to defeat the team of players.
Another game I really enjoy, and it is back out in a nice new edition, is TALISMAN, which is very easy to learn, and is really a Fantasy Lite game. It's great for occasions when you want to play something more casual, without prep. On the up side, it's got a gorgeous board, the events and creatures are ever-shifting, and everyone's fortunes rise and fall in truly riotous fashion. (Plus, people get turned into toads. Totally worth playing for the toad factor!) On the downside, this game is unusually dependent on the luck of the dice - randomness is very high and strategy is low. Also, it is ridiculously competitive, unlike the other games I mentioned. Because it is so casual, this might not matter. I just mention it because it is in the same fantasy vein, and super easy to learn and play.