  We introduce the method of desingularization of multi-variable multiple
zeta-functions (of the generalized Euler-Zagier type), under the motivation of
finding suitable rigorous meaning of the values of multiple zeta-functions at
non-positive integer points. We reveal that multiple zeta-functions (which are
known to be meromorphic in the whole space with infinitely many singular
hyperplanes) turn to be entire on the whole space after taking the
desingularization. The desingularized function is given by a suitable finite
`linear' combination of multiple zeta-functions with some arguments shifted. It
is shown that specific combinations of Bernoulli numbers attain the special
values at their non-positive integers of the desingularized ones. We also
discuss twisted multiple zeta-functions, which can be continued to entire
functions, and their special values at non-positive integer points can be
explicitly calculated.
