Claudia Kinkaid decides that she's going to run away—and that her brother Jamie will be going with her (something he doesn't know yet). In the initial set-up, we see Claudia as a particularly hard-headed little girl, though she'd hate it if anyone called her that. She clues her brother into the whole situation, and together, they come up with their daring plan.
Here's how it's going to go down: they'll escape by hiding on the school bus until everyone gets off, and then they'll take the train into New York City (a fitting place to run away to). In the exposition, it's all about planning, planning, planning—and keeping their fingers crossed that they don't get caught.
The kiddos sneak into the museum and manage to stay there after it closes by avoiding the guards and sleeping in a 16th-century bed that someone was murdered in. (Not creepy at all.) They're definitely on an adventure, but they don't know where it will lead.
When they discover that the museum has an angel statue that was purchased for $225 and may or may not have been carved by Michelangelo, they decide that this is it—their chance to be heroes. They're going to find out the statue's origins. How a couple of kids are going to do the work of a bunch of art analysts and historians is beyond us (and them), but they're ready.
Along the way, they get almost-caught a bunch of times and almost-run out of money, but they keep on keepin' on.
When they can't find much at the museum and the library, Claudia and Jamie are understandably frustrated. We mean, they've been researching the angel statue for less than a week and they still don't know whether it's a legitimate Michelangelo? Geez.
They decide to take a little trip on the train out to Connecticut, where they confront the previous owner of the statue—one Mrs. Basil E. Frankweiler—and demand that she tells them the angel's secret. Those resourceful (and kind of pushy) kids finally find the angel's secret after Mrs. Frankweiler lets them go through her oddly arranged files.
And lo and behold, the mystery is solved! The angel really was done by Michelangelo, and Claudia cries when she finds out. Not only is this super-exciting (a great attribute of a climax), it's the turning point for our protagonist. She feels like a changed little lady. And maybe, just maybe she's ready to head home.
But then, Mrs. Frankweiler drops another bomb on them—she will leave them the documents that prove the angel's origins in her will only if they agree to tell her everything about how they ran away. Of course, they agree. And so the loose ends begin to be tied.
The kids are sent back home in style—in a Rolls Royce, to be exact. In the end, the Kinkaid siblings are satisfied with their new secret (which, of course, has to do with the Italian Renaissance—usual childhood fare), and discuss how they'd like to visit Mrs. Frankweiler someday. Mrs. Frankweiler also gives the reader the big reveal that [drumroll please] Saxonberg is actually Claudia and Jamie's grandfather. What a small world, huh?