  We prove uniform boundedness statements for semistable pure sheaves on
projective manifolds. For example, we prove that the set of isomorphism classes
of pure sheaves of dimension 2 that are slope semistable with respect to ample
classes that vary in a compact set $K$ are bounded. We also prove uniform
boundedness for pure sheaves of higher dimension, but with restrictions on the
compact set $K$. As applications we get new statements about moduli spaces of
semistable sheaves, and the wall-chamber structure that governs their
variation.
