  \emph{Scalable spaces} are simply connected compact manifolds or finite
complexes whose real cohomology algebra embeds in their algebra of (flat)
differential forms. This is a rational homotopy invariant property and all
scalable spaces are formal; indeed, scalability can be thought of as a metric
version of formality. They are also characterized by particularly nice behavior
from the point of view of quantitative homotopy theory. Among other results, we
show that spaces which are formal but not scalable provide counterexamples to
Gromov's long-standing conjecture on distortion in higher homotopy groups.
