Hyperbolic metric spaces in the sense of Gromov have played an important role both in Geometric Group Theory and in Geometry and have been subject to intense research over the past 20 years. They were introduced and first studied by M. Gromov in his seminal paper on Hyperbolic Groups and can be thought of as spaces of negative curvature in a coarse sense. In this talk we will present the following optimal characterization of Gromov hyperbolicity via an isoperimetric inequality. Let X be a geodesic metric space (e.g. a Riemannian manifold) and suppose there exists epsilon > 0 such that the filling area of each sufficiently long rectifiable loop c in X is bounded above by (1 - epsilon)(4\pi)^{-1}length(c)^2. Then X is Gromov hyperbolic. (As is well-known, X then even admits a linear isoperimetric inequality, i.e. the filling area is bounded by Dlength(c) for some constant D.) Here, the filling area of c is by definition the least area of a 2-chain with boundary c. Our theorem is optimal as shows the case of Euclidean space and is new even for Riemannian manifolds. It strengthens results in Gromov's paper in which the constant (16\pi)^{-1} was obtained for a large class of Riemannian manifolds and (4000)^{-1} for geodesic metric spaces. If time permits we will give other optimal characterizations of Gromov hyperbolicity.