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.