A compact manifold with a smooth boundary is boundary rigid if its boundary and boundary distance function uniquely determine its interior up to boundary preserving isometries. Under certain natural conditions, the notion of boundary rigidity is closely related to Gromov's filling minimality. In this talk, we will first give a brief overview of Burago-Ivanov's approach to prove filling minimality and boundary rigidity for almost Euclidean and almost real hyperbolic metrics. Then we will explain how we generalize their results to regions in a rank-1 symmetric space equipped with an almost symmetric metric. We will also explain the relations to Besson-Courtois-Gallot's barycenter constructions used in their celebrated volume entropy rigidity theorem.