The Dehn function, also known as the filling area or isoperimetric function, measures how much area is needed to fill closed curves of a given length in a space by a disc-type surface. It is an important invariant used in analysis, geometry, and geometric group theory. The aim of the talk is to explore relationships between the growth of the Dehn function and the geometry of the underlying space. In particular, we study how suitable bounds on the growth of the Dehn function imply upper bounds on the curvature. The talk is based on several joint works with Alexander Lytchak and Robert Young and the methods of proof use recent results about the existence, regularity, and intrinsic structure of area minimizing discs in metric spaces.