  We give explicit formulas for the dimensions and the degrees of
$A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Our
formulas can be applied also to the case where the $A$-discriminant varieties
are higher-codimensional and their degrees are described by the geometry of the
configurations $A$. Moreover combinatorial formulas for the Euler obstructions
of general (not necessarily normal) toric varieties will be also given.
