We give upper bounds for the eigenvalues of the La-place-Beltrami operator of
a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension
and the volume of the submanifold and the order of the eigenvalue, these bounds
depend on either the maximal number of intersection points of $M$ with a
$p$-plane in a generic position (transverse to $M$), or an invariant which
measures the concentration of the volume of $M$ in $\R^{m+p}$. These bounds are
asymptotically optimal in the sense of the Weyl law. On the other hand, we show
that even for hypersurfaces (i.e., when $p=1$), the first positive eigenvalue
cannot be controlled only in terms of the volume, the dimension and (for $m\ge
3$) the differential structure.