  This is the manuscript for Proceedings of International Conference and
Workshop on Valuation Theory held at University of Saskachewan, Canada in 1999.
I have succeeded in showing that any two-dimensional hypersurface singularities
of germs of varieties in any characteristic can be resolved by iterated
monoidal transformations with centers in smooth subvarieties. The new proof for
the two-dimensional case depends on new ideas. Ideas are essentially different
from Abhyankar's one in 1956 and Lipman's one in 1978. It seems to be possible
to generalize the new proof into higher dimensional cases, if we add several
ideas further. In this article I try to explain my new ideas rather than
partial result I explained at the conference.
