From differential geometry to metric geometry and back again

Classic differential geometry, studying intrinsic properties of Riemannian manifolds and similar spaces, have developed heavy machinery to handle variuos problems. However powerful these tools are, some problems still resists them.
M. Gromov pioneered another approach: objects of differential geometry can be viewed in a context of more general metric spaces that do not have to have any differential structure (but preserve other, more geometric, features of Riemannian spaces). Furthermore, the “space” of (nice) metric spaces can be equipped with its own structure which brings new tools to defferential geometry, much like function spaces bring new tools to the study of functions.
I will survey some old and new fruits of this approach: groups of polynomial growth, curvature bounds, systolic and filling inequalities, inverse boundary problems.