  We study some of the local CP-odd and magnetic properties of the non-Abelian
vacuum with use of overlap fermions within the quenched lattice gauge theory.
Among these properties are the following: inhomogeneous spatial distribution of
the topological charge density (chirality for massless fermions) in SU(2)
gluodynamics (for uncooled gauge configurations the chirality is localized on
low-dimensional defects with d=2..3, while a sequence of cooling steps gives
rise to four-dimensional instantons and hence a four-dimensional structure of
the chirality distribution); finite local fluctuations of the chirality growing
with the strength of an external magnetic field; magnetization and
susceptibility of the QCD vacuum in SU(3) theory; magnetic catalysis of the
chiral symmetry breaking, and the electric conductivity of the QCD vacuum in
strong magnetic fields.
