  This paper summarizes the results at the present moment about singularities
with respect to the Mather-Jacobian log discrepancies over algebraically closed
field of arbitrary characteristic. The basic point is the Inversion of
Adjunction with respect to Mather-Jacobian discrepancies holds in arbitrary
characteristic. Based on this fact we will reduce many geometric properties of
the singularities into the problem on jet schemes and try to avoid discussions
which are distinctive for characteristic 0.
