This thesis has three parts. In the first part, we introduce the notions of regular and flexible Lagrangian manifolds with Legendrian boundary in Weinstein domains. We show that flexible Lagrangians satisfy an existence and uniqueness h-principle (up to ambient symplectomorphism) and give many examples of flexible Lagrangians in the standard symplectic ball. In the second part, we show that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology, generalizing similar results in the subcritical Weinstein case. We also prove relative analogs of our results for flexible Lagrangian fillings of Legendrians. As an application, we show that any closed exact, Maslov zero Lagrangian in a cotangent bundle that intersects a cotangent fiber exactly once has the same cohomology as the zero-section. In the third part, we construct many new exotic symplectic and contact structures. For instance, we show that many closed n-manifolds of dimension at least three can be realized as exact Lagrangian submanifolds of the cotangent bundle of the n-sphere with possibly an exotic symplectic structure. We also show that in dimensions at least five any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic.