In this thesis we prove some classification results for symplectic and exact Lagrangian fillings in contact geometry. First we prove a classification result for symplectic fillings of certain contact manifolds. Let ( M,xi) be a contact 3-manifold and T2 ⊂ (M,xi a mixed torus. We prove a JSJ-type decomposition theorem for strong and exact symplectic fillings of (M,xi) when (M,xi) is cut along T 2. As an application we prove the uniqueness of exact fillings when (M,xi) is obtained by Legendrian surgery on a knot in ( S3,xistd which is stabilized both positively and negatively. Second we show a classification result for Lagrangian fillings of Legendrian representatives of positive braid closures in S3. This second result follows from an injectivity result for augmentation categories of positive braids.