  We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian
manifold with boundary. The first involves the volume of the manifold and of
its boundary, as well as packing and volume growth constants of the boundary
and its distortion. Its proof is based on a metric-measure space technique that
was introduced by Colbois and Maerten. The second bound is in terms of the
extrinsic diameter of the boundary and its injectivity radius. It is obtained
from a concentration inequality, akin to Gromov-Milman concentration for closed
manifolds. By applying these bounds to cylinders over closed manifold, we
obtain bounds for eigenvalues of the Laplace operator, in the spirit of
Grigor'yan-Netrusov-Yau and of Berger-Croke. For a family of manifolds that has
uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce
that a large first nonzero Steklov eigenvalue implies that each boundary
component is contained in a ball of small extrinsic radius.
