We will discuss joint work with Bruce Kleiner and Assaf Naor. Consider a Lipschitz function, $f: R^n\to R$, with ${\rm Lip}\, f\leq 1$. By Rademacher's theorem, the differential, $f'$, exists almost everywhere. Moreover, in a quantitative sense (independent of the particular function $f$) at most locations and scales, after suitable rescaling, $f$ will be as close as one likes to some suitably chosen linear function $\ell$. However, since we assume no definite bound on the second derivatives, $f''$, it cannot be asserted that $\ell$ can always be chosen to be the first order Taylor expansion of $f$. This phenomenon, which we refer to as ``quantitative differentiation'' was first observed by Jones. We will give an abstract treatment which can be used to show that something analogous occurs in many other situations in analysis and geometric analysis, including some in which the derivative, $f'$, does not exist in a classical sense. Then we will discuss a particular such case, Lipschitz maps from the Heisenberg group to $L_1$, in which the quantitative aspect is of particular interest for applications.