  The rate at which cross sections grow with energy is sensitive to the
presence of extra dimensions in a rather model-independent fashion. We examine
how rates would be expected to grow if there are more spatial dimensions than 3
which appear at some energy scale, making connections with black hole physics
and string theory. We also review what is known about the corresponding
generalization of the Froissart-Martin bound and the experimental status of
high energy hadronic cross sections which appear to saturate it up to the
experimentally accessible limit of 100 TeV. We discuss how extra dimensions can
be searched for in high energy cross section data and find no room for large
extra dimensions in present data. Any apparent signatures of extra dimensions
at the LHC may have to be interpreted as due to some other form of new physics.
