We will present several quantitative estimates for classical min-max constructions in Riemannian geometry. Our results include upper bounds for the smallest volume of minimal hypersurfaces in terms of a conformal invariant and the volume of the ambient manifold (joint work with P. Glynn-Adey). We will also prove that if a Riemannian 2-sphere can be swept out by closed curves of length less than L then it can be sliced into simple curves of length less than L (joint work with G. R. Chambers).