So...I finished all four books. They were entertaining enough, I suppose, but I'm probably too old to be truly amused. I kept thinking of how unlikely it would be for a bunch of scientists to set aside a group of people to survive the upcoming apocalypse and then leave them without any sort of survival training whatsoever. "Let's just take care of all their needs for two hundred years or so and then kick them out. Also, we won't tell them that they live in a cave so that they won't feel bad about not having a sky, even though this leaves them pretty much ignorant of how the world works. Winter schminter, at least they won't miss having grass." Bother.
The third book ("The Prophet of Yonwood") didn't even feel like a part of the series. I felt cheated by it, since it was so indirectly about Ember. That may have frustrated me more than the gaping premise, actually.
In conclusion, I think that this series was less than fantastic. Let your kids read it, maybe read the first book, but don't feel like you're missing out if check out something else instead. Read "The Lightning Thief," by Rick Riordan. That's a good one.