  We study intersection theory and Chern classes of reflexive sheaves on normal
varieties. In particular, we define generalization of Mumford's intersection
theory on normal surfaces to higher dimensions. We also define and study the
second Chern class for reflexive sheaves on normal varieties. We use these
results to prove some Bogomolov type inequalities on normal varieties in
positive characteristic. We also prove some new boundedness results on normal
varieties in positive characteristic.
