For many classical rigidity questions in differential geometry it is natural to ask to which extent they are stable. I will review several recent results in the literature. A typical example is the following: there is a constant $C$ such that, if $\Sigma$ is a $2$-dimensional embedded closed surface in $R^3$, then $\min_\lambda \|A- \lambda g\|_{L^2} \leq C \|A - {\rm tr A} g/2\|_{L^2}$, where $A$ is the second fundamental form of the surface and $g$ the Riemannian metric as a submanifold of $R^3$.