In this talk we will address the problem of analyzing the blow-up behavior of a sequence of solutions to the Yamabe equation in high dimensions. We will show how sharp pointwise estimates can be used to prove that the Weyl tensor vanishes up to order $[\frac{n-6}{2}]$ at a blowup point. This gives a proof of the compactness of the set of solutions to the Yamabe problem, a result conjectured by R. Schoen around 15 years ago. This is joint work with Marcus Khuri and Richard Schoen.