  Let $S$ be a finitely generated standard multigraded algebra over an Artinian
local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper
answers to the question when mixed multiplicities of $M$ are positive and
characterizes them in terms of lengths of $A$-modules. As an application, we
get results on mixed multiplicities of ideals (Proposition 4.4 and Theorem
4.5), and recover the results of Risler and Teissier [Te](see Remark 4.8),
Trung and Verma [TV](see Remark 4.6).
