the (euclidean) isoperimetric inequality says that any set has larger perimeter than a ball with the same area. the quantitative isoperimetric inequality says that the difference in perimeters is bounded from below by the square of the distance from our set e to the ``closest'' ball of the same area. in this talk, we will discuss an extension of this result to closed riemannian manifolds with analytic metrics. in particular, we show that a similar inequality holds but with the distance raised to a power that depends on the geometry. we also have examples which show that a greater power than two is sometimes necessary and that the analyticity condition is necessary. this is joint work with o. chodosh (stanford) and l. spolaor (ucsd).